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Proof step 102 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))), 0 ≤ j, 0 ≤ i, 2 ^ j * 2 < i ⊦ bits = [], bits0 = [], (int2bits(bits2int(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j)))))[# bits0 - 1] ↔ ¬ bits0.last) ∧ int2bits(bits2int(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j)))))[# bits0 - 1, bits0.last] = bits0 + bits

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 103:
bits ≠ [] ⊦ int2bits(bits2int(bits)) = sgn(bits)
Proof step 104:
¬ bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))) ≠ [], # 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))), 0 ≤ j, 0 ≤ i, 2 ^ j * 2 < i ⊦ bits = [], bits0 = [], (int2bits(bits2int(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j)))))[# bits0 - 1] ↔ ¬ bits0.last) ∧ int2bits(bits2int(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j)))))[# bits0 - 1, bits0.last] = bits0 + bits
Proof step 107:
bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))) ≠ [], # 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))), 0 ≤ j, 0 ≤ i, 2 ^ j * 2 < i ⊦ bits = [], bits0 = [], (sgn(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))))[# bits0 - 1] ↔ ¬ bits0.last) ∧ sgn(bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))))[# bits0 - 1, bits0.last] = bits0 + bits

Rule argument

seq: bits ≠ [] ⊦ int2bits(bits2int(bits)) = sgn(bits) subst: [bits] → [ bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j)))] rotate: F spec: int2bits inst: name: bits2bits-def

Additional information

Used terms: bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j)))