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Proof step 16 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 ⊦ bits2bytes(sgn(bytes2bits(bytes0) + byte2bits(by))) = bytes0 + by ', b2i((bytes0 + by ').first) = ~1, b2i((bytes0 + by ').first) = 0, true ∈ bytes2bits(bytes0) ∧ false ∈ bytes2bits(bytes0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 17:
⊦ bytes ≠ [] → (bytes + bytes0).first = bytes.first
Proof step 18:
¬ bytes0 ≠ [], ∀ bytes1. # bytes1 < #(bytes0 + by ') → bytes2bits(bytes1) ≠ [] → bytes1 = [] ∨ b2i(bytes1.first) = 0 ∨ b2i(bytes1.first) = ~1 ∨ bits2bytes(sgn(bytes2bits(bytes1))) = bytes1 ⊦ bits2bytes(sgn(bytes2bits(bytes0) + byte2bits(by))) = bytes0 + by ', b2i((bytes0 + by ').first) = ~1, b2i((bytes0 + by ').first) = 0, true ∈ bytes2bits(bytes0) ∧ false ∈ bytes2bits(bytes0)
Proof step 26:
bytes0 ≠ [], ∀ bytes1. # bytes1 < #(bytes0 + by ') → bytes2bits(bytes1) ≠ [] → bytes1 = [] ∨ b2i(bytes1.first) = 0 ∨ b2i(bytes1.first) = ~1 ∨ bits2bytes(sgn(bytes2bits(bytes1))) = bytes1 ⊦ bits2bytes(sgn(bytes2bits(bytes0) + byte2bits(by))) = bytes0 + by ', b2i(bytes0.first) = ~1, b2i(bytes0.first) = 0, true ∈ bytes2bits(bytes0) ∧ false ∈ bytes2bits(bytes0)

Rule argument

seq: ⊦ bytes ≠ [] → (bytes + bytes0).first = bytes.first subst: [bytes0, bytes] → [by ', bytes0] rotate: F spec: list inst: bytelist name: first-01

Additional information

Used terms: by ', bytes0