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Proof step 8 from proof datanode-pagesize (previous step)

fo.inodes?, ro.inodes?, node-closed(fs[ri[datakey(m, n)]], ri, fs, ro), datakey(m, n) ∈ ri, inodekey(m) ∈ ri, ri[datakey(m, n)] ∈ fs, fo ⊆ dom fi, ro ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∀ adr. adr ∈ log → adr ∈ fs ∧ node-cons(fs[adr].key, fs[adr]) ⊦ # fs[ri[datakey(m, n)]].data = VFS_PAGE_SIZE, fs[ri[inodekey(m)]].directory, dups(log), inodekey(0) ∈ ri

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ 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)
Proof step 10:
ro.inodes?, fo.inodes?, ri[datakey(m, n)] ∈ fs, inodekey(m) ∈ ri, datakey(m, n) ∈ ri, ro ⊆ dom ri, fo ⊆ dom fi, fs[ri[datakey(m, n)]].dentry? → inodekey(fs[ri[datakey(m, n)]].key.ino) ∈ ri ∧ ¬ inodekey(fs[ri[datakey(m, n)]].key.ino) ∈ ro ∧ fs[ri[inodekey(fs[ri[datakey(m, n)]].key.ino)]].directory ∧ inodekey(fs[ri[datakey(m, n)]].ino) ∈ ri ∧ ¬ inodekey(fs[ri[datakey(m, n)]].ino) ∈ ro, fs[ri[datakey(m, n)]].data? → inodekey(fs[ri[datakey(m, n)]].key.ino) ∈ ri ∧ ¬ fs[ri[inodekey(fs[ri[datakey(m, n)]].key.ino)]].directory ∧ # fs[ri[datakey(m, n)]].data = VFS_PAGE_SIZE, ∀ adr. adr ∈ log → adr ∈ fs ∧ node-cons(fs[adr].key, fs[adr]), ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ # fs[ri[datakey(m, n)]].data = VFS_PAGE_SIZE, fs[ri[inodekey(m)]].directory, dups(log), inodekey(0) ∈ ri

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[datakey(m, n)]]] flags: <all paths>

Additional information

Used terms: ro, fs, ri, fs[ri[datakey(m, n)]]