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Proof step 67 from proof bits2int-pos (previous step)

bits2uint(firstn(# bits0 % 8, bits0)) ≤ 127, 7 < # bits0, # bits0 % 8 ≤ 7, # bits0 % 8 < # bits0, 0 ≤ bits2uint(firstn(# bits0 % 8, bits0)), (fillsgn(8, uint2bits(bits2uint(firstn(# bits0 % 8, bits0)))) + bytes2bits(bits2bytes(restn(# bits0 % 8, bits0)))).first ⊦ fillsgn(8, uint2bits(bits2uint(firstn(# bits0 % 8, bits0)))) + bytes2bits(bits2bytes(restn(# bits0 % 8, bits0))) = [], bits0 = [], (# bits0 + 1) % 8 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 68:
⊦ bits ≠ [] → ((bits + bits0).first ↔ bits.first)
Proof step 69:
fillsgn(8, uint2bits(bits2uint(firstn(# bits0 % 8, bits0)))) ≠ [], bits2uint(firstn(# bits0 % 8, bits0)) ≤ 127, 7 < # bits0, # bits0 % 8 ≤ 7, # bits0 % 8 < # bits0, 0 ≤ bits2uint(firstn(# bits0 % 8, bits0)), fillsgn(8, uint2bits(bits2uint(firstn(# bits0 % 8, bits0)))).first ⊦ fillsgn(8, uint2bits(bits2uint(firstn(# bits0 % 8, bits0)))) + bytes2bits(bits2bytes(restn(# bits0 % 8, bits0))) = [], bits0 = [], (# bits0 + 1) % 8 = 0

Rule argument

seq: ⊦ bits ≠ [] → ((bits + bits0).first ↔ bits.first) subst: [bits0, bits] → [bytes2bits(bits2bytes(restn(# bits0 % 8, bits0))), fillsgn(8, uint2bits(bits2uint(firstn(# bits0 % 8, bits0))))] rotate: F spec: list inst: bitlista name: first-01

Additional information

Used simplifier rules: ⊦ fillsgn(n, bits) = [] ↔ bits = [] ∧ n = 0;% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j;%0 ≤ i ⊦ uint2bits(i) ≠ []; Used terms: bytes2bits(bits2bytes(restn(# bits0 % 8, bits0))), fillsgn(8, uint2bits(bits2uint(firstn(# bits0 % 8, bits0))))