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Proof step 107 from proof int2bytes-rec-pos-any (previous step)

0 ≤ i ∧ 0 ≤ 8, # int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i) * i→n(8) + 1, i→n(i) + 1 = # bits2bytes(false ' + firstn((i→n(i) * i→n(8)) % 8, int2bits(j % 2 ^ (i * 8)).rest)) + # bits2bytes(restn((i→n(i) * i→n(8)) % 8, int2bits(j % 2 ^ (i * 8)).rest)), 2 ^ (i * 8) ≤ j, 0 < i, 0 ≤ j % 2 ^ (i * 8) ⊦ (i→n(i) * i→n(8) + 1) % 8 = 0, int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = ( bits2bytes(false ' + firstn((i→n(i) * i→n(8)) % 8, int2bits(j % 2 ^ (i * 8)).rest)) + bits2bytes(restn((i→n(i) * i→n(8)) % 8, int2bits(j % 2 ^ (i * 8)).rest)) ).rest

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 108:
# int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = 1 + 8 * i→n(i), i→n(i) + 1 = # bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) + # bits2bytes(false '), 2 ^ (i * 8) ≤ j, 0 < i, 0 ≤ j % 2 ^ (i * 8), 0 ≤ i ⊦ int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = (bits2bytes(false ') + bits2bytes(int2bits(j % 2 ^ (i * 8)).rest)).rest

Rule argument

Additional information

Used simplifier rules: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward)% ⊦ m + n = n + m;% ⊦ (m + n0) + n1 = m + n0 + n1;%n ≠ 0 ⊦ (m + n0 * n) % n = m % n;% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1;% ⊦ bits + [] = bits;% ⊦ firstn(0, bits) = [];% ⊦ restn(0, bits) = bits;% ⊦ m * n = n * m;% ⊦ m0 * m * n = (m0 * m) * n;%n ≠ 0 ⊦ (m * n) % n = 0;% ⊦ i * j = j * i;