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Proof step 356 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)) % 8 = 0, i→n(i * 8) % 8 = 0, # int2bits(j % 2 ^ (i * 8)) < i→n(i * 8), 0 < i, 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), # int2bits(j % 2 ^ (i * 8)) / 8 ≤ i→n(i), 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))) ⊦ 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, bits2bytes(mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))) = mklist(i2b(0), i→n(i) - # int2bits(j % 2 ^ (i * 8)) / 8) + bits2bytes(int2bits(j % 2 ^ (i * 8)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 357:
⊦ # bits0 % 8 = 0 ∧ bits ≠ [] ∧ bits0 ≠ [] → bits2bytes(bits + bits0) = bits2bytes(bits) + bits2bytes(bits0)
Proof step 358:
# int2bits(j % 2 ^ (i * 8)) % 8 = 0 ∧ mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) ≠ [] ∧ int2bits(j % 2 ^ (i * 8)) ≠ [], # int2bits(j % 2 ^ (i * 8)) % 8 = 0, i→n(i * 8) % 8 = 0, # int2bits(j % 2 ^ (i * 8)) < i→n(i * 8), 0 < i, 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), # int2bits(j % 2 ^ (i * 8)) / 8 ≤ i→n(i), 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))) ⊦ 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, bits2bytes(mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8)))) + bits2bytes(int2bits(j % 2 ^ (i * 8))) = mklist(i2b(0), i→n(i) - # int2bits(j % 2 ^ (i * 8)) / 8) + bits2bytes(int2bits(j % 2 ^ (i * 8)))

Rule argument

name: bits2bytes-app rotate: F subst: [bits0, bits] → [int2bits(j % 2 ^ (i * 8)), mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8)))]

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ mklist(boolvar, m) = [] ↔ m = 0;% ⊦ n < m → m - n ≠ 0;% ⊦ int2bits(j) ≠ []; Used terms: int2bits(j % 2 ^ (i * 8)), mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8)))