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Proof step 5 from proof keys-empty (previous step)

∀ key. ¬(∃ n. key = inodekey(n) ∧ n ∈ ns) ⊦ # ns = 0, ns = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ ns = ns0 ↔ (∀ n. n ∈ ns ↔ n ∈ ns0)
Proof step 7:
∀ key. ¬(∃ n. key = inodekey(n) ∧ n ∈ ns) ⊦ # ns = 0, ∀ n. n ∈ ns ↔ n ∈ ∅

Rule argument

spec: set-basic, inst: nat-set name: Extension seq: ⊦ ns = ns0 ↔ (∀ n. n ∈ ns ↔ n ∈ ns0) subst: [ns0, ns] → [∅, ns] flags: <all paths>

Additional information

Used terms: ∅, ns