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Proof step 14 from proof shiftzero-rec-neg-shorter (previous step)

i < ~1, # int2bits(i) ≤ n, i < 0, 1 < j, 0 < j ⊦ bits2int(firstn(n, mklist(false, i→n(j)) + fillsgn(n, int2bits(i)))) = bits2int(firstn(# fillsgn(n - 1, int2bits((i - 1) / 2)), mklist(false, i→n(j) - 1) + fillsgn(n - 1, int2bits((i - 1) / 2))))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ # bits ≤ n → # fillsgn(n, bits) = n
Proof step 16:
¬ # int2bits((i - 1) / 2) ≤ n - 1, i < ~1, # int2bits(i) ≤ n, i < 0, 1 < j, 0 < j ⊦ bits2int(firstn(n, mklist(false, i→n(j)) + fillsgn(n, int2bits(i)))) = bits2int(firstn(# fillsgn(n - 1, int2bits((i - 1) / 2)), mklist(false, i→n(j) - 1) + fillsgn(n - 1, int2bits((i - 1) / 2))))
Proof step 19:
# int2bits((i - 1) / 2) ≤ n - 1, i < ~1, # int2bits(i) ≤ n, i < 0, 1 < j, 0 < j ⊦ bits2int(firstn(n, mklist(false, i→n(j)) + fillsgn(n, int2bits(i)))) = bits2int(firstn(n - 1, mklist(false, i→n(j) - 1) + fillsgn(n - 1, int2bits((i - 1) / 2))))

Rule argument

seq: ⊦ # bits ≤ n → # fillsgn(n, bits) = n subst: [bits, n] → [int2bits((i - 1) / 2), n - 1] rotate: F spec: bitopsaux inst: name: fill-length

Additional information

Used terms: int2bits((i - 1) / 2), n - 1