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Proof step 3 from proof datalkeys-drop-orphans-in (previous step)

ro.inodes?, key ∈ ro ⊦ # datakeys(key, n, ri ⊖ ro) = 0, # ro = 0, datakeys(key, n, ri ⊖ ro) = ∅, ro = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ ks = ks0 ↔ (∀ key. key ∈ ks ↔ key ∈ ks0)

Rule argument

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

Additional information

Used simplifier rules: ⊦ key ∈ datakeys(key0, n, ri) ↔ key ∈ datakeys(key0, ri) ∧ n ≤ key.part * VFS_PAGE_SIZE; (s)%⊦ key ∈ datakeys(key0, ri) ↔ key ∈ ri ∧ key.data? ∧ key0.inode? ∧ key.ino = key0.ino; (s)% ⊦ ¬ key ∈ ∅; (s)%⊦ key ∈ ri ⊖ ks ↔ key ∈ ri ∧ ¬ key ∈ ks ∧ (key.data? → ¬ inodekey(key.ino) ∈ ks); (ws)% ⊦ key.inode? → inodekey(key.ino) = key; (s) Used terms: ∅, datakeys(key, n, ri ⊖ ro)