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Proof step 14 from proof new-inode-exists (previous step)

key ∈ ks, ∀ ks0. # ks0 < # ks → true → (∃ ino. ∀ ino0. inodekey(ino0) ∈ ks0 → ino0 < ino) ⊦ # ks = 0, ∃ ino. ∀ ino0. inodekey(ino0) ∈ ks → ino0 < ino

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ key ∈ ks ↔ (∃ ks0. ks = ks0 ++ key ∧ ¬ key ∈ ks0)
Proof step 16:
∃ ks0. ks = ks0 ++ key ∧ ¬ key ∈ ks0, ∀ ks0. # ks0 < # ks → true → (∃ ino. ∀ ino0. inodekey(ino0) ∈ ks0 → ino0 < ino) ⊦ # ks = 0, ∃ ino. ∀ ino0. inodekey(ino0) ∈ ks → ino0 < ino

Rule argument

spec: set-basic, inst: node-index name: indef seq: ⊦ key ∈ ks ↔ (∃ ks0. ks = ks0 ++ key ∧ ¬ key ∈ ks0) subst: [ks, key] → [ks, key] flags: <all paths>

Additional information

Used terms: ks, key