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Proof step 13 from proof bytes2bytes-other (previous step)

∀ bytes1. # bytes1 < #(bytes0 + by ') → bytes2bits(bytes1) ≠ [] → bytes1 = [] ∨ b2i(bytes1.first) = 0 ∨ b2i(bytes1.first) = ~1 ∨ bits2bytes(sgn(bytes2bits(bytes1))) = bytes1 ⊦ b2i((bytes0 + by ').first) = 0, b2i((bytes0 + by ').first) = ~1, bits2bytes(sgn(bytes2bits(bytes0) + byte2bits(by))) = bytes0 + by '

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ true ∈ bits ∧ false ∈ bits → sgn(bits + bits0) = sgn(bits) + bits0
Proof step 15:
¬(true ∈ bytes2bits(bytes0) ∧ false ∈ bytes2bits(bytes0)), ∀ bytes1. # bytes1 < #(bytes0 + by ') → bytes2bits(bytes1) ≠ [] → bytes1 = [] ∨ b2i(bytes1.first) = 0 ∨ b2i(bytes1.first) = ~1 ∨ bits2bytes(sgn(bytes2bits(bytes1))) = bytes1 ⊦ b2i((bytes0 + by ').first) = 0, b2i((bytes0 + by ').first) = ~1, bits2bytes(sgn(bytes2bits(bytes0) + byte2bits(by))) = bytes0 + by '
Proof step 39:
true ∈ bytes2bits(bytes0) ∧ false ∈ bytes2bits(bytes0), ∀ bytes1. # bytes1 < #(bytes0 + by ') → bytes2bits(bytes1) ≠ [] → bytes1 = [] ∨ b2i(bytes1.first) = 0 ∨ b2i(bytes1.first) = ~1 ∨ bits2bytes(sgn(bytes2bits(bytes1))) = bytes1 ⊦ b2i((bytes0 + by ').first) = 0, b2i((bytes0 + by ').first) = ~1, bits2bytes(sgn(bytes2bits(bytes0)) + byte2bits(by)) = bytes0 + by '

Rule argument

seq: ⊦ true ∈ bits ∧ false ∈ bits → sgn(bits + bits0) = sgn(bits) + bits0 subst: [bits0, bits] → [byte2bits(by), bytes2bits(bytes0)] rotate: F spec: bitopsaux inst: name: sgn-append-01

Additional information

Used terms: byte2bits(by), bytes2bits(bytes0)