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Proof step 9 from proof ri-closed-gc-one (previous step)

node-closed(fs[ri[key]], ri, fs, ro), ri[key] ∈ fs, key ∈ ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ adr ∈ fs, ( fs[ri[key]].dentry? → inodekey(fs[ri[key]].key.ino) ∈ ri[key, adr] ∧ ¬ inodekey(fs[ri[key]].key.ino) ∈ ro ∧ fs[adr, fs[ri[key]]][ri[key, adr][inodekey(fs[ri[key]].key.ino)]].directory ∧ inodekey(fs[ri[key]].ino) ∈ ri[key, adr] ∧ ¬ inodekey(fs[ri[key]].ino) ∈ ro) ∧ ( fs[ri[key]].data? → inodekey(fs[ri[key]].key.ino) ∈ ri[key, adr] ∧ ¬ fs[adr, fs[ri[key]]][ri[key, adr][inodekey(fs[ri[key]].key.ino)]].directory ∧ # fs[ri[key]].data = VFS_PAGE_SIZE)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: node-closed seq: ⊦ node-closed(nd, ri, fs, ro) ↔ ( nd.dentry? → inodekey(nd.key.ino) ∈ ri ∧ ¬ inodekey(nd.key.ino) ∈ ro ∧ fs[ri[inodekey(nd.key.ino)]].directory ∧ inodekey(nd.ino) ∈ ri ∧ ¬ inodekey(nd.ino) ∈ ro) ∧ ( nd.data? → inodekey(nd.key.ino) ∈ ri ∧ ¬ fs[ri[inodekey(nd.key.ino)]].directory ∧ # nd.data = VFS_PAGE_SIZE) subst: [ro, fs, ri, nd] → [ro, fs, ri, fs[ri[key]]] flags: <all paths>

Additional information

Used terms: ro, fs, ri, fs[ri[key]]