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Proof step 237 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)) = (i→n(i) - 1) * 8 + n, 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))), 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8, 0 < i ⊦ n = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], 1 + 8 * i→n(i) = (i→n(i) - 1) * 8 + n, (i→n(i) - 1) * 8 + n = 8 * i→n(i), i = 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 238:
⊦ # bits % 8 = 0 ∧ 8 < # bits → bits2bytes(bits) = bits2bytes(firstn(8, bits)) + bits2bytes(restn(8, bits))
Proof step 239:
¬(#(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)) = (i→n(i) - 1) * 8 + n, 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))), 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8, 0 < i ⊦ n = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], 1 + 8 * i→n(i) = (i→n(i) - 1) * 8 + n, (i→n(i) - 1) * 8 + n = 8 * i→n(i), i = 0, bits2bytes(mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8))) = bits2bytes(int2bits(j % 2 ^ (i * 8)))
Proof step 256:
#(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)) = (i→n(i) - 1) * 8 + n, 0 ≤ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))), 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8, 0 < i ⊦ n = 0, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], 1 + 8 * i→n(i) = (i→n(i) - 1) * 8 + n, (i→n(i) - 1) * 8 + n = 8 * i→n(i), i = 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))