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Proof step 110 from proof int2bytes-rec-pos-any (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 111:
# int2bits(j % 2 ^ (i * 8)).rest % 8 = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8) + 1, i→n(i) + 1 = # bits2bytes(true ' + firstn(i→n(i * 8) % 8, int2bits(j % 2 ^ (i * 8)).rest)) + # bits2bytes(restn(i→n(i * 8) % 8, int2bits(j % 2 ^ (i * 8)).rest)), 2 ^ (i * 8) ≤ j, 0 < i, j % 2 ^ (i * 8) < 0 ⊦ (i→n(i * 8) + 1) % 8 = 0, int2bits(j % 2 ^ (i * 8)).rest = [], bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = (bits2bytes(true ' + firstn(i→n(i * 8) % 8, int2bits(j % 2 ^ (i * 8)).rest)) + bits2bytes(restn(i→n(i * 8) % 8, int2bits(j % 2 ^ (i * 8)).rest)) ).rest

Rule argument

Additional information

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