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Proof step 13 from proof new-inode-exists (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
key ∈ ks, ∀ ks0. # ks0 < # ks → true → (∃ ino. ∀ ino0. inodekey(ino0) ∈ ks0 → ino0 < ino) ⊦ # ks = 0, ∃ ino. ∀ ino0. inodekey(ino0) ∈ ks → ino0 < ino

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ key ∈ ∅; (s)