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Proof step 135 from proof bxor-flip-one-bit (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 136:
⊦ (bits.first ↔ ¬ bits.rest.first) ∧ 1 < # bits → sgn(bits) = bits
Proof step 137:
¬ ( ( ((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j)))).first ↔ ¬ ((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j)))).rest.first) ∧ 1 < #((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j))))), int2bits(i) = (bits1 + boolvar ') + bits, # bits = i→n(j), bitsxor(int2bits(i), (false ' + true ') + mklist(false, i→n(j))) = (bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j))), 2 ^ j * 2 < i, 0 ≤ i, 0 ≤ j ⊦ bits = [], bits1 = [], (sgn((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j))))[# bits1] ↔ ¬ boolvar) ∧ sgn((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j))))[# bits1, boolvar] = int2bits(i)
Proof step 157:
( ((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j)))).first ↔ ¬ ((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j)))).rest.first) ∧ 1 < #((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j)))), int2bits(i) = (bits1 + boolvar ') + bits, # bits = i→n(j), bitsxor(int2bits(i), (false ' + true ') + mklist(false, i→n(j))) = (bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j))), 2 ^ j * 2 < i, 0 ≤ i, 0 ≤ j ⊦ bits = [], bits1 = [], (((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j))))[# bits1] ↔ ¬ boolvar) ∧ ((bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j))))[# bits1, boolvar] = int2bits(i)

Rule argument

seq: ⊦ (bits.first ↔ ¬ bits.rest.first) ∧ 1 < # bits → sgn(bits) = bits subst: [bits] → [ (bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j)))] rotate: F spec: bitopsaux inst: name: sgn-base-05

Additional information

Used terms: (bits1 + (¬ boolvar)') + bitsxor(bits, mklist(false, i→n(j)))