[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 114 from proof bxor-flip-one-bit (previous step)

# bits = i→n(j), int2bits(i) = bits0 + bits, bitsxor(bits0 + bits, (false ' + true ') + mklist(false, i→n(j))) = bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))), bits0 = bits0.first ', 0 ≤ j, 0 ≤ i, 2 ^ j * 2 < i ⊦ bits = [], bits0 = [], bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))) = [], (sgn(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))))[0] ↔ ¬ bits0.first) ∧ sgn(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))))[0, bits0.first] = bits0 + bits

User applied rule insert elim lemma

And obtained the following new goals

Proof step 115:
bits0 = boolvar + bits1, # bits = i→n(j), int2bits(i) = bits0 + bits, bitsxor(bits0 + bits, (false ' + true ') + mklist(false, i→n(j))) = bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))), bits0 = boolvar ', 0 ≤ j, 0 ≤ i, 2 ^ j * 2 < i ⊦ bits = [], bits0 = [], bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))) = [], (sgn(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))))[0] ↔ ¬ boolvar) ∧ sgn(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))))[0, boolvar] = bits0 + bits

Rule argument

seq: ⊦ bits ≠ [] → ( (boolvar ↔ bits.first) ∧ bits0 = bits.rest ↔ bits = boolvar + bits0) subst: [bits, boolvar, bits0] → [bits0, boolvar, bits1] rotate: T spec: list-data inst: bitlista name: elim-1

Additional information

Used simplifier rules: ⊦ bits ≠ [] → ( (boolvar ↔ bits.first) ∧ bits0 = bits.rest ↔ bits = boolvar + bits0); (elim) Used terms: bits0, boolvar, bits1