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Proof step 102 from proof bits2bits-sgn (previous step)

# lastn(8, bits) = 8, firstn(# bits - 8, bits) + lastn(8, bits) = bits, b2i(i2b(bits2int(lastn(8, bits)))) < 0, bits2int(lastn(8, bits)) < 0, 8 < # bits, ~128 ≤ bits2int(lastn(8, bits)), ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ sgn(lastn(8, bits)) = sgn(firstn(# bits - 8, bits) + lastn(8, bits)), bits = [], false ∈ bytes2bits(bits2bytes(firstn(# bits - 8, bits))), false ∈ firstn(# bits - 8, bits)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 103:
⊦ ¬ false ∈ bits ∧ bits ≠ [] → sgn(bits + bits0) = sgn(true ' + bits0)
Proof step 104:
¬ false ∈ firstn(# bits - 8, bits) ∧ firstn(# bits - 8, bits) ≠ [], # lastn(8, bits) = 8, firstn(# bits - 8, bits) + lastn(8, bits) = bits, b2i(i2b(bits2int(lastn(8, bits)))) < 0, bits2int(lastn(8, bits)) < 0, 8 < # bits, ~128 ≤ bits2int(lastn(8, bits)), ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ sgn(lastn(8, bits)) = sgn(true ' + lastn(8, bits)), bits = [], false ∈ bytes2bits(bits2bytes(firstn(# bits - 8, bits))), false ∈ firstn(# bits - 8, bits)

Rule argument

seq: ⊦ ¬ false ∈ bits ∧ bits ≠ [] → sgn(bits + bits0) = sgn(true ' + bits0) subst: [bits0, bits] → [lastn(8, bits), firstn(# bits - 8, bits)] rotate: F spec: bitopsaux inst: name: sgn-append-05

Additional information

Used simplifier rules: ⊦ firstn(n, bits) = [] ↔ n = 0;% ⊦ n < m → m - n ≠ 0; Used terms: lastn(8, bits), firstn(# bits - 8, bits)