[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 176 from proof int2bytes-rec-pos-any (previous step)

i→n(i * 8) % 8 = 0, 0 ≤ j % 2 ^ (i * 8), 2 ^ (i * 8) ≤ j, # int2bits(j % 2 ^ (i * 8)) < i→n(i * 8), 8 < # int2bits(j % 2 ^ (i * 8)), b2i (( 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)))) ).first) < 0, 1 < i, 0 < i ⊦ # int2bits(j % 2 ^ (i * 8)) % 8 = 0, i = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8), i→n(i * 8) + 1 = # int2bits(j % 2 ^ (i * 8)), mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8)) = [], bits2bytes(mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))) = fix (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))))) .rest

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 177:
⊦ bytes ≠ [] → (bytes + bytes0).first = bytes.first
Proof step 178:
bits2bytes(firstn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))) ≠ [], i→n(i * 8) % 8 = 0, 0 ≤ j % 2 ^ (i * 8), 2 ^ (i * 8) ≤ j, # int2bits(j % 2 ^ (i * 8)) < i→n(i * 8), 8 < # int2bits(j % 2 ^ (i * 8)), b2i(bits2bytes(firstn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))).first) < 0, 1 < i, 0 < i ⊦ # int2bits(j % 2 ^ (i * 8)) % 8 = 0, i = 0, # int2bits(j % 2 ^ (i * 8)) = i→n(i * 8), i→n(i * 8) + 1 = # int2bits(j % 2 ^ (i * 8)), mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8)) = [], bits2bytes(mklist(false, i→n(i * 8) - # int2bits(j % 2 ^ (i * 8))) + int2bits(j % 2 ^ (i * 8))) = fix (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))))) .rest

Rule argument

seq: ⊦ bytes ≠ [] → (bytes + bytes0).first = bytes.first subst: [bytes0, bytes] → [bits2bytes (restn (# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))), bits2bytes (firstn (# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8))))] rotate: F spec: list inst: bytelist name: first-01

Additional information

Used simplifier rules: bits2bytes-empty: ⊦ bits2bytes(bits) ≠ [];% ⊦ i * j = j * i; Used terms: bits2bytes (restn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8)))), bits2bytes (firstn(# int2bits(j % 2 ^ (i * 8)) % 8, int2bits(j % 2 ^ (i * 8))))