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Proof step 3 from proof payload-keys (previous step)

key ∈ payload(zbrar, n, nused), nused + n ≤ # zbrar ⊦ key ∈ keys(zbrar, n, nused)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
m + nused ≤ # zbrar ⊦ key ∈ payload(zbrar, m, nused) ↔ (∃ n. n < nused ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key)
Proof step 5:
nused + n ≤ # zbrar, n + nused ≤ # zbrar, ∃ n0. n0 < nused ∧ zbrar[n + n0].entry? ∧ zbrar[n + n0].key = key ⊦ key ∈ keys(zbrar, n, nused)

Rule argument

spec: <toplevel> name: in-payload seq: m + nused ≤ # zbrar ⊦ key ∈ payload(zbrar, m, nused) ↔ (∃ n. n < nused ∧ zbrar[m + n].entry? ∧ zbrar[m + n].key = key) subst: [nused, m, zbrar, key] → [nused, n, zbrar, key] flags: <all paths>

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic Used terms: nused, n, zbrar, key