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Proof step 111 from proof refine-evict (previous step)

# links(m, cache_dirs) = 0, ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[m] = mkfile(m1, md1, 0, content0), cache_files[m] = mkfile(m0, md0, 0, content), 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), m ∈ cache_files, cache_sync → synced(cache_files, cache_icache, cache_pcache), ∀ key. key ∈ cache_pcache → key.data? ⊦ m = ROOT_INO, m ∈ cache_dirs, ∀ ino. (ino ∈ ((cache_files ⊕ cache_pcache) ⊕ cache_icache) -- m ↔ ino ∈ ((cache_files -- m) ⊕ evict(m, cache_pcache)) ⊕ (cache_icache -- m)) ∧ ( ino ∈ ((cache_files ⊕ cache_pcache) ⊕ cache_icache) -- m → (((cache_files ⊕ cache_pcache) ⊕ cache_icache) -- m)[ino] = (((cache_files -- m) ⊕ evict(m, cache_pcache)) ⊕ (cache_icache -- m))[ino])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 112:
# links(m, cache_dirs) = 0, cache_files[m] = mkfile(m0, md0, 0, content), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[m] = mkfile(m1, md1, 0, content0), 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), m ∈ cache_files, ino ∈ cache_files, cache_sync → synced(cache_files, cache_icache, cache_pcache), ∀ key. key ∈ cache_pcache → key.data? ⊦ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[ino] = (((cache_files -- m) ⊕ evict(m, cache_pcache)) ⊕ (cache_icache -- m))[ino], ino = m, m = ROOT_INO, m ∈ cache_dirs

Rule argument

Additional information

Used simplifier rules: ino ∈ files ⊦ ino ≠ ino0 → (files -- ino0)[ino] = files[ino]; (s)% ⊦ ino ∈ files ⊕ pcache ↔ ino ∈ files; (s)% ⊦ ino ∈ files ⊕ icache ↔ ino ∈ files; (s)% ⊦ ino ∈ files -- ino0 ↔ ino ≠ ino0 ∧ ino ∈ files; (s)