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Proof step 104 from proof bits2bits-sgn (previous step)

¬ 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)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 105:
~128 ≤ i, i ≤ 127 ⊦ b2i(i2b((* i))) = i
Proof step 106:
~128 ≤ bits2int(lastn(8, bits)) ∧ bits2int(lastn(8, bits)) ≤ 127, ¬ false ∈ firstn(# bits - 8, bits) ∧ firstn(# bits - 8, bits) ≠ [], # lastn(8, bits) = 8, firstn(# bits - 8, bits) + lastn(8, bits) = bits, 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: ~128 ≤ i, i ≤ 127 ⊦ b2i(i2b((* i))) = i subst: [i] → [bits2int(lastn(8, bits))] rotate: F spec: byte inst: name: num-ibbi

Additional information

Used simplifier rules: ⊦ firstn(n, bits) = [] ↔ n = 0;% ⊦ n < m → m - n ≠ 0;%i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j;% ⊦ ¬ i ≤ j ↔ j < i;% ⊦ 8 < # bits → ¬ 127 < bits2int(lastn(8, bits)); Used terms: bits2int(lastn(8, bits))