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Proof step 5 from proof keys-inodekey-in (previous step)

∀ ns0, n. # ns0 < # ns → true → (inodekey(n) ∈ ns0.keys ↔ n ∈ ns0) ⊦ # ns = 0, inodekey(n) ∈ ns.keys ↔ n ∈ ns, 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:
∀ ns0, n. # ns0 < # ns → true → (inodekey(n) ∈ ns0.keys ↔ n ∈ ns0) ⊦ # ns = 0, inodekey(n) ∈ ns.keys ↔ n ∈ ns, ∀ 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