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Proof step 10 from proof keys-mod (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
n ≠ 0, m + n + nused ≤ # zbrar ⊦ keys(zbrar[m, zbr], m + n, nused) = keys(zbrar, m + n, nused)

Rule argument

spec: <toplevel> name: keys-drop-lower seq: n ≠ 0, m + n + nused ≤ # zbrar ⊦ keys(zbrar[m, zbr], m + n, nused) = keys(zbrar, m + n, nused) subst: [nused, n, zbr, m, zbrar] → [nused0, 1, mkzentry(zbrar[m].key, adr), m, zbrar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: nused0, 1, mkzentry(zbrar[m].key, adr), m, zbrar