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Proof step 86 from proof bits2bits-sgn (previous step)

~128 ≤ bits2int(lastn(8, bits)), 8 < # bits, bits2int(lastn(8, bits)) < 0, ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ bits = [], sgn(true ' + byte2bits(i2b(bits2int(lastn(8, bits))))) = sgn(bits), false ∈ firstn(# bits - 8, bits), false ∈ bytes2bits(bits2bytes(firstn(# bits - 8, bits)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 87:
bits ≠ [], bits.first ↔ boolvar ⊦ sgn(boolvar ' + bits) = sgn(bits)
Proof step 88:
¬(byte2bits(i2b(bits2int(lastn(8, bits)))) ≠ [] ∧ (byte2bits(i2b(bits2int(lastn(8, bits)))).first ↔ true)), ~128 ≤ bits2int(lastn(8, bits)), 8 < # bits, bits2int(lastn(8, bits)) < 0, ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ bits = [], sgn(true ' + byte2bits(i2b(bits2int(lastn(8, bits))))) = sgn(bits), false ∈ firstn(# bits - 8, bits), false ∈ bytes2bits(bits2bytes(firstn(# bits - 8, bits)))
Proof step 91:
byte2bits(i2b(bits2int(lastn(8, bits)))) ≠ [] ∧ (byte2bits(i2b(bits2int(lastn(8, bits)))).first ↔ true), ~128 ≤ bits2int(lastn(8, bits)), 8 < # bits, bits2int(lastn(8, bits)) < 0, ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ bits = [], sgn(byte2bits(i2b(bits2int(lastn(8, bits))))) = sgn(bits), false ∈ firstn(# bits - 8, bits), false ∈ bytes2bits(bits2bytes(firstn(# bits - 8, bits)))

Rule argument

seq: bits ≠ [], bits.first ↔ boolvar ⊦ sgn(boolvar ' + bits) = sgn(bits) subst: [bits, boolvar] → [byte2bits(i2b(bits2int(lastn(8, bits)))), true] rotate: F spec: bitopsaux inst: name: sgn-rec-01

Additional information

Used terms: byte2bits(i2b(bits2int(lastn(8, bits)))), true