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Proof step 12 from proof keys-mod (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
n0 < nused0, m + nused0 < # 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, nused0 + 1) = keys(zbrar, m, nused0 + 1) ++ zbrar[m + n0 + 1].key

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m +1 = m + 1; (s)