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Proof step 9 from proof keys-mod (previous step)

m + nused0 < # zbrar, m < # zbrar[m, mkzentry(zbrar[m].key, adr)], sorted(zbrar, m, nused0 + 1), ∀ n, zbrar, m, adr. n < nused0 ∧ sorted(zbrar, m, nused0) ∧ m + nused0 ≤ # zbrar → keys(zbrar[m + n, mkzentry(zbrar[m + n].key, adr)], m, nused0) = keys(zbrar, m, nused0) ++ zbrar[m + n].key ⊦ keys(zbrar[m, mkzentry(zbrar[m].key, adr)], m + 1, nused0) ++ zbrar[m, mkzentry(zbrar[m].key, adr)][m].key = keys(zbrar, m, nused0 + 1) ++ zbrar[m].key

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
m + nused0 < # zbrar, m < # zbrar, sorted(zbrar, m, nused0 + 1), ∀ n, zbrar, m, adr. n < nused0 ∧ sorted(zbrar, m, nused0) ∧ m + nused0 ≤ # zbrar → keys(zbrar[m + n, mkzentry(zbrar[m + n].key, adr)], m, nused0) = keys(zbrar, m, nused0) ++ zbrar[m + n].key ⊦ keys(zbrar[m, mkzentry(zbrar[m].key, adr)], m + 1, nused0) ++ zbrar[m].key = keys(zbrar, m + 1, nused0) ++ zbrar[m].key

Rule argument

Additional information

Used simplifier rules: keys-rec: ⊦ m < # zbrar → keys(zbrar, m, nused + 1) = keys(zbrar, m + 1, nused) ++ zbrar[m].key;% ⊦ ks ++ key ++ key = ks ++ key; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ mkzentry(key, adr).key = key; (s)%m < # zbrar ⊦ zbrar[m, zbr][m] = zbr; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ # zbrar[n, zbr] = # zbrar; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)