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Proof step 60 from proof bits2bits-sgn (previous step)

firstn(# bits - 8, bits) + lastn(8, bits) = bits, 8 < # bits, b2i(i2b(bits2int(lastn(8, bits)))) < 0, ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ bits = [], false ' + byte2bits(i2b(bits2int(lastn(8, bits)))) = sgn(false ' + lastn(8, bits)), true ∈ bytes2bits(bits2bytes(firstn(# bits - 8, bits))), true ∈ firstn(# bits - 8, bits)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 61:
⊦ bits ≠ [] ∧ (bits.first ↔ ¬ boolvar) → sgn(boolvar ' + bits) = boolvar ' + bits
Proof step 62:
¬(lastn(8, bits) ≠ [] ∧ (lastn(8, bits).first ↔ ¬ false)), firstn(# bits - 8, bits) + lastn(8, bits) = bits, 8 < # bits, b2i(i2b(bits2int(lastn(8, bits)))) < 0, ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ bits = [], false ' + byte2bits(i2b(bits2int(lastn(8, bits)))) = sgn(false ' + lastn(8, bits)), true ∈ bytes2bits(bits2bytes(firstn(# bits - 8, bits))), true ∈ firstn(# bits - 8, bits)
Proof step 65:
lastn(8, bits) ≠ [] ∧ (lastn(8, bits).first ↔ ¬ false), firstn(# bits - 8, bits) + lastn(8, bits) = bits, 8 < # bits, b2i(i2b(bits2int(lastn(8, bits)))) < 0, ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ bits = [], false ' + byte2bits(i2b(bits2int(lastn(8, bits)))) = false ' + lastn(8, bits), true ∈ bytes2bits(bits2bytes(firstn(# bits - 8, bits))), true ∈ firstn(# bits - 8, bits)

Rule argument

seq: ⊦ bits ≠ [] ∧ (bits.first ↔ ¬ boolvar) → sgn(boolvar ' + bits) = boolvar ' + bits subst: [bits, boolvar] → [lastn(8, bits), false] rotate: F spec: bitopsaux inst: name: sgn-base-03

Additional information

Used terms: lastn(8, bits), false