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Proof step 182 from proof int2bytes-rec-neg-05 (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 183:
⊦ # bits % 8 ≠ 0 ∧ 8 < # bits → bits2bytes(bits) = bits2bytes(firstn(# bits % 8, bits)) + bits2bytes(restn(# bits % 8, bits))
Proof step 184:
¬(# int2bits(j % 2 ^ (i * 8)) % 8 ≠ 0 ∧ 8 < # int2bits(j % 2 ^ (i * 8))), i→n(i) + 1 = # bits2bytes(int2bits(j % 2 ^ (i * 8))), 1 + 8 * i→n(i) = # int2bits(j % 2 ^ (i * 8)), 0 ≤ i, j % 2 ^ (i * 8) < 0, j + 2 ^ (i * 8) < i, j + 2 ^ (i * 8) ≤ 0, j + 2 ^ (i * 8 - 1) < 0, 0 < i ⊦ bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = bits2bytes(int2bits(j % 2 ^ (i * 8))).rest, j % 2 ^ (i * 8) = 0, int2bits(j % 2 ^ (i * 8)).rest = []
Proof step 191:
# int2bits(j % 2 ^ (i * 8)) % 8 ≠ 0 ∧ 8 < # int2bits(j % 2 ^ (i * 8)), i→n(i) + 1 = # ( 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))))), 1 + 8 * i→n(i) = # int2bits(j % 2 ^ (i * 8)), 0 ≤ i, j % 2 ^ (i * 8) < 0, j + 2 ^ (i * 8) < i, j + 2 ^ (i * 8) ≤ 0, j + 2 ^ (i * 8 - 1) < 0, 0 < i ⊦ bits2bytes(int2bits(j % 2 ^ (i * 8)).rest) = ( 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)))) ).rest, j % 2 ^ (i * 8) = 0, int2bits(j % 2 ^ (i * 8)).rest = []

Rule argument

name: bits2bytes-rec-01 rotate: F subst: [bits] → [int2bits(j % 2 ^ (i * 8))]

Additional information

Used terms: int2bits(j % 2 ^ (i * 8))