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Proof step 108 from proof bxor-flip-one-bit (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 109:
⊦ 1 < # bits ∧ 1 < # bits0 → bitsxor(bits, bits0) = bitsxor(bits.butlast, bits0.butlast) + (¬(bits.last ↔ bits0.last))
Proof step 110:
¬(1 < # bits0 ∧ 1 < #(false ' + true ')), bitsxor(bits0 + bits, (false ' + true ') + mklist(false, i→n(j))) = bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))), int2bits(i) = bits0 + bits, # bits = i→n(j), 2 ^ j * 2 < i, 0 ≤ i, 0 ≤ j ⊦ bitsxor(bits0, false ' + true ') + bitsxor(bits, mklist(false, i→n(j))) = [], bits0 = [], bits = [], (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
Proof step 129:
1 < # bits0 ∧ 1 < #(false ' + true '), bitsxor(bits0 + bits, (false ' + true ') + mklist(false, i→n(j))) = (bitsxor(bits0.butlast, (false ' + true ').butlast) + (¬(bits0.last ↔ (false ' + true ').last))) + bitsxor(bits, mklist(false, i→n(j))), int2bits(i) = bits0 + bits, # bits = i→n(j), 2 ^ j * 2 < i, 0 ≤ i, 0 ≤ j ⊦ (bitsxor(bits0.butlast, (false ' + true ').butlast) + (¬(bits0.last ↔ (false ' + true ').last))) + bitsxor(bits, mklist(false, i→n(j))) = [], bits0 = [], bits = [], ( sgn((bitsxor(bits0.butlast, (false ' + true ').butlast) + (¬(bits0.last ↔ (false ' + true ').last))) + bitsxor(bits, mklist(false, i→n(j)))) [# bits0 - 1 ] ↔ ¬ bits0.last) ∧ sgn((bitsxor(bits0.butlast, (false ' + true ').butlast) + (¬(bits0.last ↔ (false ' + true ').last))) + bitsxor(bits, mklist(false, i→n(j)))) [# bits0 - 1, bits0.last ] = bits0 + bits

Rule argument

name: bitsxor-rec-01 rotate: F subst: [bits0, bits] → [false ' + true ', bits0]

Additional information

Used terms: false ' + true ', bits0