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Proof step 15 from proof keys-mod (previous step)

m + nused0 < # zbrar, n0 < nused0, m < # zbrar[m + n0 + 1, mkzentry(zbrar[m + n0 + 1].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 + n0 + 1, mkzentry(zbrar[m + n0 + 1].key, adr)], m + 1, nused0) ++ zbrar[m + n0 + 1, mkzentry(zbrar[m + n0 + 1].key, adr)][m].key = keys(zbrar, m, nused0 + 1) ++ zbrar[m + n0 + 1].key

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
n0 < nused0, 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 + n0 + 1, mkzentry(zbrar[m + n0 + 1].key, adr)], m + 1, nused0) ++ zbrar[m].key = keys(zbrar, m + 1, nused0) ++ zbrar[m].key ++ zbrar[m + n0 + 1].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;% ⊦ # zbrar[n, zbr] = # zbrar; (s)%m ≠ n ⊦ zbrar[m, zbr][n] = zbrar[n]; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ m = m + n ↔ n = 0; (s)