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Proof step 65 from proof refine-rename-dir-negdentry (previous step)

# links(m3, cache_dirs) = m4, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries[str], cache_dirs) = cache_dirs[entries[str]].nlink, cache_dirs[entries[str]] = mkdir(cache_dirs[entries[str]].size, md5, cache_dirs[entries[str]].nlink, cache_dirs[entries[str]].nsubdirs, entries2), cache_dirs[m] = mkdir(cache_dirs[m].size, md4, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), cache_dirs[m3] = mkdir(m0 + 1, md2, m4, m1 + 1, entries), entries[str] = entries0[str], cache_dirs[m3] = mkdir(m2, md3, m4, m5, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, ((cache_dirs[m3].entries:= (entries0 -- str)).size:= m0).nsubdirs:= m1] [m, ( ( (m = m3 ⊃ ((cache_dirs[m3].entries:= (entries0 -- str)).size:= m0).nsubdirs:= m1;cache_dirs[m]) .entries:= (m = m3 ⊃ ((cache_dirs[m3].entries:= (entries0 -- str)).size:= m0).nsubdirs:= m1;cache_dirs[m]).entries [str0, entries[str] ]) .size:= ((m = m3 ⊃ ((cache_dirs[m3].entries:= (entries0 -- str)).size:= m0).nsubdirs:= m1;cache_dirs[m]).size + 1)) .nsubdirs:= ((m = m3 ⊃ ((cache_dirs[m3].entries:= (entries0 -- str)).size:= m0).nsubdirs:= m1;cache_dirs[m]).nsubdirs + 1) ], m2 = m0 + 1, m5 = m1 + 1, 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), entries[str] ∈ cache_dirs, str ∈ entries, m3 ∈ cache_dirs, m ∈ cache_dirs, str ∈ entries0, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_inode_sync#(mkinode(m3, md0, true, m4, m1 + 1, m0 + 1); dirs, cache_files); afs_inode_sync#(mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, cache_files); new_child_inode := new_child_inode0; skip; old_parent_inode := (m3 ∈ dirs ⊃ mkinode(m3, dirs[m3].meta, true, dirs[m3].nlink, dirs[m3].nsubdirs, dirs[m3].size) ;mkinode(m3, cache_files[m3].meta, false, cache_files[m3].nlink, 0, cache_files[m3].size) ); new_parent_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) ); err := ESUCCESS; asynced := ?; new_dent := mkdentry(str0, mkdentry(str, entries[str]).ino); old_dent1 := negdentry(mkdentry(str, entries[str]).name); if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, cache_icache); icache_set#(new_parent_inode, asynced; err, cache_icache); skip; dcache_delete#(old_parent_inode.ino, old_dent1.name; err, cache_dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, cache_dcache)}] ( icache-cons(cache_icache, dirs, cache_files) ∧ dcache-cons(cache_dcache, dirs, cache_files) ∧ pcache-cons(cache_pcache, dirs, cache_files) ∧ afs-cons(dirs, cache_files) ∧ synced(dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); old_parent_inode0 := mkinode(m3, md0, true, # links(m3, cache_dirs), m1 + 1, m0 + 1); old_child_inode0 := mkinode(entries[str], md1, true, # links(entries[str], cache_dirs), cache_dirs[entries[str]].nsubdirs, cache_dirs[entries[str]].size); cache_dirs0 := cache_dirs; cache_dirs1 := cache_dirs; cache_dirs2 := cache_dirs; afs_rename# (; old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode0 ∧ new_parent_inode = new_parent_inode0 ∧ mkinode (entries[str], md1, true, # links(entries[str], cache_dirs0), cache_dirs1[entries[str]].nsubdirs, cache_dirs2[entries[str]].size) = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent1 = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced))), str0 ∈ cache_dirs[m].entries, str0 ∈ entries1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 66:
# links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[m3] = mkdir(m0 + 1, md3, # links(m3, cache_dirs), m1 + 1, entries0), cache_dirs[m] = mkdir(cache_dirs[m].size, md4, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md5, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, entries2), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, mkdir(m0, md3, # links(m3, cache_dirs), m1, entries0 -- str)] [m, ( ( (m = m3 ⊃ mkdir(m0, md3, # links(m3, cache_dirs), m1, entries0 -- str);cache_dirs[m]) .entries:= (m = m3 ⊃ mkdir(m0, md3, # links(m3, cache_dirs), m1, entries0 -- str);cache_dirs[m]).entries [str0, entries0[str] ]) .size:= ((m = m3 ⊃ mkdir(m0, md3, # links(m3, cache_dirs), m1, entries0 -- str);cache_dirs[m]).size + 1)) .nsubdirs:= ((m = m3 ⊃ mkdir(m0, md3, # links(m3, cache_dirs), m1, entries0 -- str);cache_dirs[m]).nsubdirs + 1) ], 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, m3 ∈ cache_dirs, str ∈ entries0, entries0[str] ∈ cache_dirs, ¬ str0 ∈ entries1, ¬ str0 ∈ cache_dirs[m].entries, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ [afs_inode_sync#(mkinode(m3, md0, true, # links(m3, cache_dirs), m1 + 1, m0 + 1); dirs, cache_files); afs_inode_sync#(mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, cache_files); new_child_inode := new_child_inode0; skip; old_parent_inode := (m3 ∈ dirs ⊃ mkinode(m3, dirs[m3].meta, true, dirs[m3].nlink, dirs[m3].nsubdirs, dirs[m3].size) ;mkinode(m3, cache_files[m3].meta, false, cache_files[m3].nlink, 0, cache_files[m3].size) ); new_parent_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) ); err := ESUCCESS; asynced := ?; new_dent := mkdentry(str0, mkdentry(str, entries0[str]).ino); old_dent1 := negdentry(mkdentry(str, entries0[str]).name); if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, cache_icache); icache_set#(new_parent_inode, asynced; err, cache_icache); skip; dcache_delete#(old_parent_inode.ino, old_dent1.name; err, cache_dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, cache_dcache)}] ( icache-cons(cache_icache, dirs, cache_files) ∧ dcache-cons(cache_dcache, dirs, cache_files) ∧ pcache-cons(cache_pcache, dirs, cache_files) ∧ afs-cons(dirs, cache_files) ∧ synced(dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨old_dent0 := mkdentry(str, entries0[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); old_parent_inode0 := mkinode(m3, md0, true, # links(m3, cache_dirs), m1 + 1, m0 + 1); old_child_inode0 := mkinode (entries0[str], md1, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size); cache_dirs0 := cache_dirs; cache_dirs1 := cache_dirs; cache_dirs2 := cache_dirs; afs_rename# (; old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode0 ∧ new_parent_inode = new_parent_inode0 ∧ mkinode (entries0[str], md1, true, # links(entries0[str], cache_dirs0), cache_dirs1[entries0[str]].nsubdirs, cache_dirs2[entries0[str]].size) = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent1 = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced)))

Rule argument

Additional information

Used simplifier rules: ⊦ mkdir(m, md, m0, m1, entries).nsubdirs:= m4 = mkdir(m, md, m0, m4, entries); (s)%⊦ mkdir(m, md, m0, m1, entries).size:= m2 = mkdir(m2, md, m0, m1, entries); (s)%⊦ mkdir(m, md, m0, m1, entries).entries:= entries0 = mkdir(m, md, m0, m1, entries0); (s)%⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)%afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base