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Proof step 296 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)) = m * 8 + n, i→n(i * 8) % 8 = 0, n < 8, m < i→n(i), 0 ≤ j % 2 ^ (i * 8), 2 ^ (i * 8) ≤ j, 0 < i, m * 8 + n < i→n(i * 8), 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))) ⊦ i = 0, m * 8 + n = i→n(i * 8), i→n(i * 8) + 1 = m * 8 + n, mklist(false, 8 * (i→n(i) - m + 1)) + mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i) - m + 1 = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], n = 0, bits2bytes(mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))) = bits2bytes(int2bits(j % 2 ^ (i * 8)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 297:
⊦ # bits % 8 = 0 ∧ 8 < # bits → bits2bytes(bits) = bits2bytes(firstn(8, bits)) + bits2bytes(restn(8, bits))
Proof step 298:
¬(#(mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))) % 8 = 0 ∧ 8 < #(mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)))), # int2bits(j % 2 ^ (i * 8)) = m * 8 + n, i→n(i * 8) % 8 = 0, n < 8, m < i→n(i), 0 ≤ j % 2 ^ (i * 8), 2 ^ (i * 8) ≤ j, 0 < i, m * 8 + n < i→n(i * 8), 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))) ⊦ i = 0, m * 8 + n = i→n(i * 8), i→n(i * 8) + 1 = m * 8 + n, mklist(false, 8 * (i→n(i) - m + 1)) + mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i) - m + 1 = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], n = 0, bits2bytes(mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))) = bits2bytes(int2bits(j % 2 ^ (i * 8)))
Proof step 313:
#(mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))) % 8 = 0 ∧ 8 < #(mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))), # int2bits(j % 2 ^ (i * 8)) = m * 8 + n, i→n(i * 8) % 8 = 0, n < 8, m < i→n(i), 0 ≤ j % 2 ^ (i * 8), 2 ^ (i * 8) ≤ j, 0 < i, m * 8 + n < i→n(i * 8), 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))) ⊦ i = 0, m * 8 + n = i→n(i * 8), i→n(i * 8) + 1 = m * 8 + n, mklist(false, 8 * (i→n(i) - m + 1)) + mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i) - m + 1 = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], n = 0, bits2bytes(firstn(8, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)))) + bits2bytes(restn(8, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)))) = bits2bytes(int2bits(j % 2 ^ (i * 8)))

Rule argument

name: bits2bytes-rec-02 rotate: F subst: [bits] → [mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))]

Additional information

Used terms: mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))