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Proof step 92 from proof bits2bits-sgn (previous step)

~128 ≤ bits2int(lastn(8, bits)), 8 < # bits, bits2int(lastn(8, bits)) < 0, b2i(i2b(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)))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 93:
⊦ # bits = 8 → byte2bits(i2b(bits2int(bits))) = bits
Proof step 94:
# lastn(8, bits) ≠ 8, ~128 ≤ bits2int(lastn(8, bits)), 8 < # bits, bits2int(lastn(8, bits)) < 0, b2i(i2b(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)))
Proof step 97:
# lastn(8, bits) = 8, ~128 ≤ bits2int(lastn(8, bits)), 8 < # bits, bits2int(lastn(8, bits)) < 0, b2i(i2b(bits2int(lastn(8, bits)))) < 0, ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ bits = [], sgn(lastn(8, bits)) = sgn(bits), false ∈ firstn(# bits - 8, bits), false ∈ bytes2bits(bits2bytes(firstn(# bits - 8, bits)))

Rule argument

name: byte2bits8 rotate: F subst: [bits] → [lastn(8, bits)]

Additional information

Used terms: lastn(8, bits)