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Proof step 3 from proof keys-mod-0 (previous step)

n < nused, n < # zbrar, sorted(zbrar, 0, nused), nused ≤ # zbrar ⊦ keys(zbrar[n, mkzentry(zbrar[n].key, adr)], 0, nused) = keys(zbrar, 0, nused) ++ zbrar[n].key

User applied rule apply lemma

And obtained the following new goals

Proof step 4:
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

Rule argument

spec: <toplevel> name: keys-mod seq: 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 subst: [n, nused, m, zbrar, key, adr] → [n, nused, 0, zbrar, zbrar[n].key, adr]

Additional information

Used simplifier rules: ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: n, nused, 0, zbrar, zbrar[n].key, adr