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Proof step 112 from proof refine-evict (previous step)

# 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

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 113:
⊦ ino ∈ icache ∧ ¬ icache[ino].inode.directory → (files ⊕ icache)[ino] = mkfile(icache[ino].inode.size, icache[ino].inode.meta, icache[ino].inode.nlink, files[ino].content)
Proof step 114:
# 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), ino ∈ cache_files, m ∈ 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, ino ∈ cache_icache -- m ∧ ¬ (cache_icache -- m)[ino].inode.directory
Proof step 316:
# 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), ino ∈ cache_files, m ∈ cache_files, ino ∈ cache_icache -- m, cache_sync → synced(cache_files, cache_icache, cache_pcache), ∀ key. key ∈ cache_pcache → key.data? ⊦ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[ino] = mkfile ((cache_icache -- m)[ino].inode.size, (cache_icache -- m)[ino].inode.meta, (cache_icache -- m)[ino].inode.nlink, ((cache_files -- m) ⊕ evict(m, cache_pcache))[ino].content), ino = m, m = ROOT_INO, (cache_icache -- m)[ino].inode.directory, m ∈ cache_dirs

Rule argument

spec: cache-base name: at-icache-overwrite seq: ⊦ ino ∈ icache ∧ ¬ icache[ino].inode.directory → (files ⊕ icache)[ino] = mkfile (icache[ino].inode.size, icache[ino].inode.meta, icache[ino].inode.nlink, files[ino].content) subst: [ino, icache, files] → [ino, cache_icache -- m, (cache_files -- m) ⊕ evict(m, cache_pcache)] flags: <all paths>

Additional information

Used terms: ino, cache_icache -- m, (cache_files -- m) ⊕ evict(m, cache_pcache)