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Proof step 73 from proof shiftzero-pos (previous step)

i→n(j) < # int2bits(i), 0 ≤ i, # int2bits(i) < n, i→n(j) < n, i→n(j) + 1 < n ⊦ # int2bits(i) = i→n(j), bits2int(mklist(false, i→n(j)) + firstn(n - i→n(j), mklist(false, n - # int2bits(i)) + int2bits(i))) = bits2int(firstn(# int2bits(i) - i→n(j), int2bits(i))), firstn(# int2bits(i) - i→n(j), int2bits(i)).first

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 74:
⊦ # bits ≤ n → firstn(n, bits + bits0) = bits + firstn(n - # bits, bits0)
Proof step 75:
¬ # mklist(false, n - # int2bits(i)) ≤ n - i→n(j), i→n(j) < # int2bits(i), 0 ≤ i, # int2bits(i) < n, i→n(j) < n, i→n(j) + 1 < n ⊦ # int2bits(i) = i→n(j), bits2int(mklist(false, i→n(j)) + firstn(n - i→n(j), mklist(false, n - # int2bits(i)) + int2bits(i))) = bits2int(firstn(# int2bits(i) - i→n(j), int2bits(i))), firstn(# int2bits(i) - i→n(j), int2bits(i)).first
Proof step 77:
# mklist(false, n - # int2bits(i)) ≤ n - i→n(j), i→n(j) < # int2bits(i), 0 ≤ i, # int2bits(i) < n, i→n(j) < n, i→n(j) + 1 < n ⊦ # int2bits(i) = i→n(j), bits2int(mklist(false, i→n(j)) + mklist(false, n - # int2bits(i)) + firstn(n - i→n(j) - # mklist(false, n - # int2bits(i)), int2bits(i))) = bits2int(firstn(# int2bits(i) - i→n(j), int2bits(i))), firstn(# int2bits(i) - i→n(j), int2bits(i)).first

Rule argument

seq: ⊦ # bits ≤ n → firstn(n, bits + bits0) = bits + firstn(n - # bits, bits0) subst: [bits0, bits, n] → [int2bits(i), mklist(false, n - # int2bits(i)), n - i→n(j)] rotate: F spec: list-del inst: bitlista name: firstn-app-01

Additional information

Used terms: int2bits(i), mklist(false, n - # int2bits(i)), n - i→n(j)