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Proof step 2 from proof keys-empty (previous step)

ns.keys = ∅ ⊦ # ns = 0, ns = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ ks = ks0 ↔ (∀ key. key ∈ ks ↔ key ∈ ks0)
Proof step 4:
∀ key. key ∈ ns.keys ↔ key ∈ ∅ ⊦ # ns = 0, ns = ∅

Rule argument

spec: set-basic, inst: node-index name: Extension seq: ⊦ ks = ks0 ↔ (∀ key. key ∈ ks ↔ key ∈ ks0) subst: [ks0, ks] → [∅, ns.keys] flags: <all paths>

Additional information

Used terms: ∅, ns.keys