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Proof step 262 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)) % 8 ≠ 0 ∧ 8 < # int2bits(j % 2 ^ (i * 8)), # int2bits(j % 2 ^ (i * 8)) = (i→n(i) - 1) * 8 + n, n < 8 + (i→n(i) - 1) * 8, n < (i→n(i) - 1) * 8 + 7, 0 ≤ bytes2int ( 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))))), 2 ^ (i * 8) ≤ j, 0 ≤ j % 2 ^ (i * 8), n < 8, 0 < i ⊦ i→n(i) - 1 = 0, 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(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))))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 263:
# int2bits(j % 2 ^ (i * 8)) = (i→n(i) - 1) * 8 + n, 0 < i, n < 8, 0 ≤ j % 2 ^ (i * 8), 2 ^ (i * 8) ≤ j, n < (i→n(i) - 1) * 8 + 7, n < 8 + (i→n(i) - 1) * 8, 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))))) ⊦ i = 0, (i→n(i) - 1) * 8 + n = 8 * i→n(i), 1 + 8 * i→n(i) = (i→n(i) - 1) * 8 + n, mklist(false, 8 - n) + int2bits(j % 2 ^ (i * 8)) = [], i→n(i) - 1 = 0, 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(firstn(n, int2bits(j % 2 ^ (i * 8)))) + bits2bytes(restn(n, int2bits(j % 2 ^ (i * 8)))), (i→n(i) - 1) * 8 = 1

Rule argument

Additional information

Used simplifier rules: ⊦ m < m + n ↔ n ≠ 0;% ⊦ m0 * m = 0 ↔ m0 = 0 ∨ m = 0;% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1);% ⊦ m + n < m + n0 ↔ n < n0;% ⊦ m +1 = m + 1;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m0 * m * n = (m0 * m) * n;% ⊦ m < n → m % n = m;%n ≠ 0 ⊦ (m + n0 * n) % n = m % n;% ⊦ m < n → n ≠ 0;% ⊦ m < n → n ≠ 0; (forward)% ⊦ m < n ∧ n = n0 → m < n0; (forward)% ⊦ m + n = n + m;% ⊦ m * n = n * m;% ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward)% ⊦ n < n0 ∧ n0 < n1 → n +1 < n1; (forward)% ⊦ i * j = j * i;