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Proof step 269 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)) = (i→n(i) - 1) * 8 + n, 0 ≤ bytes2int(bits2bytes(firstn(n, int2bits(j % 2 ^ (i * 8)))) + bits2bytes(restn(n, int2bits(j % 2 ^ (i * 8))))), 8 < (i→n(i) - 1) * 8 + n, n < 8 + (i→n(i) - 1) * 8, n < (i→n(i) - 1) * 8 + 7, 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8, 0 < i ⊦ n = 0, i→n(i) - 1 = 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, (i→n(i) - 1) * 8 = 1, bits2bytes(firstn(8, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)))) = bits2bytes(firstn(n, int2bits(j % 2 ^ (i * 8))))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 270:
⊦ # bits ≤ 8 → bits2bytes(bits) = i2b(bits2int(bits)) '
Proof step 271:
# firstn(n, int2bits(j % 2 ^ (i * 8))) ≤ 8, # int2bits(j % 2 ^ (i * 8)) = (i→n(i) - 1) * 8 + n, 0 ≤ bytes2int(i2b(bits2int(firstn(n, int2bits(j % 2 ^ (i * 8))))) ' + bits2bytes(restn(n, int2bits(j % 2 ^ (i * 8))))), 8 < (i→n(i) - 1) * 8 + n, n < 8 + (i→n(i) - 1) * 8, n < (i→n(i) - 1) * 8 + 7, 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8, 0 < i ⊦ n = 0, i→n(i) - 1 = 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, (i→n(i) - 1) * 8 = 1, bits2bytes(firstn(8, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)))) = i2b(bits2int(firstn(n, int2bits(j % 2 ^ (i * 8))))) '

Rule argument

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

Additional information

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