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Proof step 9 from proof keys-inodekeys (previous step)

n ∈ ns, ∀ ns0. # ns0 < # ns → true → ns0.keys.inodes? ⊦ # ns = 0, ns.keys.inodes?

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ n ∈ ns ↔ (∃ ns0. ns = ns0 ++ n ∧ ¬ n ∈ ns0)
Proof step 11:
∃ ns0. ns = ns0 ++ n ∧ ¬ n ∈ ns0, ∀ ns0. # ns0 < # ns → true → ns0.keys.inodes? ⊦ # ns = 0, ns.keys.inodes?

Rule argument

spec: set-basic, inst: nat-set name: indef seq: ⊦ n ∈ ns ↔ (∃ ns0. ns = ns0 ++ n ∧ ¬ n ∈ ns0) subst: [ns, n] → [ns, n] flags: <all paths>

Additional information

Used terms: ns, n