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Proof step 42 from proof bytes2bytes-zero (previous step)

8 = # fillsgn(8, int2bits(b2i(by0))), b2i(by) = 0, b2i(by0) < 0, ∀ by0, by, bytes0. # bytes0 < # [] → b2i(by) = 0 ∧ b2i(by0) < 0 → bits2bytes(sgn(byte2bits(by) + byte2bits(by0) + bytes2bits(bytes0))) = by ' + by0 ' + bytes0 ⊦ fillsgn(8, int2bits(b2i(by0))) = [], bits2bytes(firstn(1, false ' + fillsgn(8, int2bits(b2i(by0))))) = by '

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 43:
⊦ bits ≠ [] → firstn(1, bits) = bits.first '
Proof step 44:
false ' + fillsgn(8, int2bits(b2i(by0))) ≠ [], 8 = # fillsgn(8, int2bits(b2i(by0))), b2i(by) = 0, b2i(by0) < 0, ∀ by0, by, bytes0. # bytes0 < # [] → b2i(by) = 0 ∧ b2i(by0) < 0 → bits2bytes(sgn(byte2bits(by) + byte2bits(by0) + bytes2bits(bytes0))) = by ' + by0 ' + bytes0 ⊦ fillsgn(8, int2bits(b2i(by0))) = [], bits2bytes((false ' + fillsgn(8, int2bits(b2i(by0)))).first ') = by '

Rule argument

seq: ⊦ bits ≠ [] → firstn(1, bits) = bits.first ' subst: [bits] → [false ' + fillsgn(8, int2bits(b2i(by0)))] rotate: F spec: list-del inst: bitlista name: firstN-one

Additional information

Used simplifier rules: ⊦ ¬ bits + bits0 ≠ [] ↔ bits = [] ∧ bits0 = [];% ⊦ boolvar ' ≠ [];% ⊦ (bits + bits0) + bits1 = bits + bits0 + bits1; Used terms: false ' + fillsgn(8, int2bits(b2i(by0)))