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Proof step 41 from proof refine-evict (previous step)

# links(m, cache_dirs) = 0, cache_dirs[m] = mkdir(m0, md0, 0, m2, entries), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), 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), ino ∈ cache_files, m ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache) ⊦ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[ino] = ((cache_files ⊕ cache_pcache) ⊕ (cache_icache -- m))[ino], m = ROOT_INO, ino ∈ cache_icache -- m ∧ ¬ (cache_icache -- m)[ino].inode.directory

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 42:
# links(m, cache_dirs) = 0, cache_dirs[m] = mkdir(m0, md0, 0, m2, entries), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), 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), ino ∈ cache_files, m ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache) ⊦ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[ino] = ((cache_files ⊕ cache_pcache) ⊕ (cache_icache -- m))[ino], ino = m, m = ROOT_INO, ¬ cache_icache[ino].inode.directory ∧ ino ∈ cache_icache

Rule argument

Additional information

Used simplifier rules: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs → ¬ ino ∈ files; (s)% ⊦ ino ≠ ino0 → (icache -- ino0)[ino] = icache[ino]; (s)% ⊦ ino ∈ icache -- ino0 ↔ ino ≠ ino0 ∧ ino ∈ icache; (s)