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Proof step 29 from proof refine-lookup (previous step)

cache_dirs[p_ino] = mkdir(m, md, # links(p_ino, cache_dirs), m1, entries), cache_dcache[dentrykey(p_ino, dent0.name)] ≠ mkdentry(dent0.name, entries[dent0.name]), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), dent0.name ∈ entries, dentrykey(p_ino, dent0.name) ∈ cache_dcache, p_ino ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(p_ino, cache_dirs) ≠ ∅ ∨ p_ino = ROOT_INO ⊦ ∃ err0. ⌊err0⌋ ∧ ⟨err := err0⟩ (cache_dcache[dentrykey(p_ino, dent0.name)] = dent0 ∧ ESUCCESS = err)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 30:
⊦ dcache-cons(dcache, dirs, files) ↔ (∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ valid-dentry(key.ino, dcache[key], dirs, files))
Proof step 31:
cache_dirs[p_ino] = mkdir(m, md, # links(p_ino, cache_dirs), m1, entries), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), p_ino ∈ cache_dirs, dentrykey(p_ino, dent0.name) ∈ cache_dcache, dent0.name ∈ entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), ∀ key. key ∈ cache_dcache → key.dentry? ∧ key.name = cache_dcache[key].name ∧ valid-dentry(key.ino, cache_dcache[key], cache_dirs, cache_files), links(p_ino, cache_dirs) ≠ ∅ ∨ p_ino = ROOT_INO ⊦ ∃ err0. ⌊err0⌋ ∧ ⟨err := err0⟩ (cache_dcache[dentrykey(p_ino, dent0.name)] = dent0 ∧ ESUCCESS = err), cache_dcache[dentrykey(p_ino, dent0.name)] = mkdentry(dent0.name, entries[dent0.name])

Rule argument

spec: cache-base name: dcache-cons seq: ⊦ dcache-cons(dcache, dirs, files) ↔ (∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ valid-dentry(key.ino, dcache[key], dirs, files)) subst: [files, dirs, dcache] → [cache_files, cache_dirs, cache_dcache] flags: <all paths>

Additional information

Used terms: cache_files, cache_dirs, cache_dcache