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Proof step 1 from proof keys-mod (conclusion)

n < nused, m + nused ≤ # zbrar, zbrar[m + n].key = key, sorted(zbrar, m, nused) ⊦ keys(zbrar[m + n, mkzentry(key, adr)], m, nused) = keys(zbrar, m, nused) ++ key

System applied rule normalize

And obtained the following new goals

Proof step 2:
zbrar[m + n].key = key, n < nused, sorted(zbrar, m, nused), m + nused ≤ # zbrar ⊦ keys(zbrar[m + n, mkzentry(key, adr)], m, nused) = keys(zbrar, m, nused) ++ key

Proof step 1 from proof keys-mod (conclusion)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information