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Proof step 19 from proof refine-mkdir (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[m] = mkdir(cache_dirs[m].size, md2, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), dent = negdentry(str), dirs = cache_dirs, err = err0, ino ≠ 0, 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_dirs, ¬ str ∈ cache_dirs[m].entries, ¬ str ∈ entries0, ¬ ino ∈ cache_files, ¬ ino ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_mkdir_ino#(m, md, ino; dent, err, dirs); cache_files0 := cache_files; afs_inode_sync#(mkinode(m, md0, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, cache_files); p_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, cache_files[m].meta, false, cache_files[m].nlink, 0, cache_files[m].size) ); c_inode := getinode(dent.ino, dirs, cache_files); asynced := ?; afs_files := (cache_files0 ⊕ cache_pcache) ⊕ cache_icache; if err = ESUCCESS then let asynced = false in {icache_set#(p_inode, asynced; err, cache_icache); icache_set#(c_inode, asynced; err, cache_icache); dcache_set#(p_inode.ino, dent.name, dent; err, cache_dcache)}⟫ ⟨dent0 := negdentry(str); p_inode0 := mkinode(m, md0, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); afs_mkdir#(md; p_inode0, p_inode1, dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 20:
# links(m, cache_dirs) = m1, cache_dirs[m] = mkdir(m0, md2, m1, m2, entries0), cache_dirs[m] = mkdir(m0, md1, m1, m2, entries), dent = negdentry(str), dirs = cache_dirs, err = err0, 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_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_mkdir_ino#(m, md, ino; dent, err, dirs); cache_files0 := cache_files; afs_inode_sync#(mkinode(m, md0, true, # links(m, cache_dirs), m2, m0); dirs, cache_files); p_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, cache_files[m].meta, false, cache_files[m].nlink, 0, cache_files[m].size) ); c_inode := getinode(dent.ino, dirs, cache_files); asynced := ?; afs_files := (cache_files0 ⊕ cache_pcache) ⊕ cache_icache; if err = ESUCCESS then let asynced = false in {icache_set#(p_inode, asynced; err, cache_icache); icache_set#(c_inode, asynced; err, cache_icache); dcache_set#(p_inode.ino, dent.name, dent; err, cache_dcache)}⟫ ⟨dent0 := negdentry(str); p_inode0 := mkinode(m, md0, true, # links(m, cache_dirs), m2, m0); afs_mkdir#(md; p_inode0, p_inode1, dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0), ino = 0, str ∈ entries, str ∈ entries0, ino ∈ cache_files, ino ∈ cache_dirs

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [cache_dirs[m], m0, md1, m1, m2, entries]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: cache_dirs[m], m0, md1, m1, m2, entries