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Proof step 125 from proof int2bytes-rec-pos-any (previous step)

# int2bits(j % 2 ^ (i * 8)) = i→n(i * 8), i→n(i * 8) % 8 = 0, bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))) < 0, # bits2bytes(int2bits(j % 2 ^ (i * 8))) ≤ i→n(i), 2 ^ (i * 8) ≤ j, 0 < i, 0 ≤ j % 2 ^ (i * 8) ⊦ bits2bytes(int2bits(j % 2 ^ (i * 8))) = fix(i→n(i) + 1, bits2bytes(int2bits(j % 2 ^ (i * 8)))).rest

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 126:
⊦ # bytes < n ∧ bytes2int(bytes) < 0 → fix(n, bytes) = mklist(i2b(~1), n - # bytes) + bytes
Proof step 127:
# bits2bytes(int2bits(j % 2 ^ (i * 8))) < i→n(i) + 1 ∧ bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))) < 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8), i→n(i * 8) % 8 = 0, bytes2int(bits2bytes(int2bits(j % 2 ^ (i * 8)))) < 0, # bits2bytes(int2bits(j % 2 ^ (i * 8))) ≤ i→n(i), 2 ^ (i * 8) ≤ j, 0 < i, 0 ≤ j % 2 ^ (i * 8) ⊦ bits2bytes(int2bits(j % 2 ^ (i * 8))) = (mklist(i2b(~1), i→n(i) + 1 - # bits2bytes(int2bits(j % 2 ^ (i * 8)))) + bits2bytes(int2bits(j % 2 ^ (i * 8)))).rest

Rule argument

name: fix-neg rotate: F subst: [bytes, n] → [bits2bytes(int2bits(j % 2 ^ (i * 8))), i→n(i) + 1]

Additional information

Used simplifier rules: ⊦ i * j = j * i;% ⊦ m < n + 1 ↔ ¬ n < m;% ⊦ m ≤ n → (n < m ↔ false);% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m;% ⊦ ¬ i < j ↔ j ≤ i;% ⊦ i < j → (¬ ¬ j ≤ i ↔ false); Used terms: bits2bytes(int2bits(j % 2 ^ (i * 8))), i→n(i) + 1