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Proof step 11 from proof new-inode-exists (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ ks = ks0 ↔ (∀ key. key ∈ ks ↔ key ∈ ks0)
Proof step 13:
∀ ks0. # ks0 < # ks → true → (∃ ino. ∀ ino0. inodekey(ino0) ∈ ks0 → ino0 < ino) ⊦ # ks = 0, ∀ key. key ∈ ks ↔ key ∈ ∅, ∃ ino. ∀ ino0. inodekey(ino0) ∈ ks → ino0 < ino

Rule argument

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

Additional information

Used terms: ∅, ks