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Proof step 69 from proof bits2bits-sgn (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 70:
~128 ≤ i, i ≤ 127 ⊦ b2i(i2b((* i))) = i
Proof step 71:
~128 ≤ bits2int(lastn(8, bits)) ∧ bits2int(lastn(8, bits)) ≤ 127, firstn(# bits - 8, bits) + lastn(8, bits) = bits, lastn(8, bits).first, bits2int(lastn(8, bits)) < 0, 8 < # bits, ∀ bits0. # bits0 < # bits → true → bits0 = [] ∨ sgn(bytes2bits(bits2bytes(bits0))) = sgn(bits0) ⊦ # lastn(8, bits) = 8, byte2bits(i2b(bits2int(lastn(8, bits)))) = lastn(8, bits), bits = [], true ∈ firstn(# bits - 8, bits), true ∈ bytes2bits(bits2bytes(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: ⊦ ¬ i ≤ j ↔ j < i;% ⊦ 8 < # bits → ¬ bits2int(lastn(8, bits)) < ~128;% ⊦ 8 < # bits → ¬ 127 < bits2int(lastn(8, bits)); Used terms: bits2int(lastn(8, bits))