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Proof step 59 from proof refine-rename-dir-negdentry (previous step)

# links(m3, cache_dirs) = m9, # links(entries[str], cache_dirs) = m1, # links(m, cache_dirs) = m12, asynced0 ↔ old_ino = new_ino ∧ new_dent = old_dent, asynced1 ↔ new_dent.dentry?, cache_dirs[m3] = mkdir(m5, md2, # links(m3, cache_dirs), m11, entries), cache_dirs[m] = mkdir(m2, md4, # links(m, cache_dirs), m10, entries1), cache_dirs[entries[str]] = mkdir(m4, md5, # links(entries[str], cache_dirs), m8, entries2), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, ¬ asynced0, ¬ asynced1, dirs = dirs4[new_ino, dirs4[new_ino].nsubdirs:= (dirs4[new_ino].nsubdirs + 1)], dirs0 = cache_dirs, dirs1 = dirs0[old_ino, dirs0[old_ino].entries:= (dirs0[old_ino].entries -- old_dent.name)], dirs2 = dirs1[old_ino, dirs1[old_ino].size:= (dirs1[old_ino].size - 1)], dirs3 = dirs2[old_ino, dirs2[old_ino].nsubdirs:= (dirs2[old_ino].nsubdirs - 1)] [new_ino, dirs2[old_ino, dirs2[old_ino].nsubdirs:= (dirs2[old_ino].nsubdirs - 1)][new_ino] .entries:= dirs2[old_ino, dirs2[old_ino].nsubdirs:= (dirs2[old_ino].nsubdirs - 1)][new_ino].entries[new_dent.name, old_dent.ino] ], dirs4 = dirs3[new_ino, dirs3[new_ino].size:= (dirs3[new_ino].size + 1)], err = err0, files = cache_files, new_child_inode = new_child_inode0, new_dent = negdentry(str0), new_ino = new_parent_inode.ino, new_parent_inode = mkinode(m, md, true, # links(m, cache_dirs), m10, m2), old_child_inode = mkinode(entries[str], md1, true, # links(entries[str], cache_dirs), m8, m4), old_dent = mkdentry(str, entries[str]), old_ino = old_parent_inode.ino, old_parent_inode = mkinode(m3, md0, true, # links(m3, cache_dirs), m11, m5), old_dent.dentry?, 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), pre-rename(old_parent_inode, old_dent, old_child_inode, new_parent_inode, new_dent, new_child_inode, dirs0, files), synced(cache_dirs, cache_icache, cache_pcache), old_dent.ino ∈ dirs2, m ∈ cache_dirs, m3 ∈ cache_dirs, entries[str] ∈ cache_dirs, str ∈ entries, old_dent.ino ∈ dirs4, dirs1[old_ino].size ≥ 1, dirs2[old_ino].nsubdirs ≥ 1, 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#(old_parent_inode; dirs, files); afs_inode_sync#(new_parent_inode; dirs, files); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, files); new_child_inode := getinode(new_child_inode.ino, dirs, files)}; old_parent_inode := getinode(old_parent_inode.ino, dirs, files); new_parent_inode := getinode(new_parent_inode.ino, dirs, files); err := ESUCCESS; asynced := ?; new_dent := mkdentry(new_dent.name, old_dent.ino); old_dent1 := negdentry(old_dent.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, files) ∧ dcache-cons(cache_dcache, dirs, files) ∧ pcache-cons(cache_pcache, dirs, files) ∧ afs-cons(dirs, files) ∧ synced(dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(files, cache_icache, cache_pcache)) → ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m, md, true, m12, m10, m2); old_parent_inode0 := mkinode(m3, md0, true, m9, m11, m5); old_child_inode0 := mkinode(entries[str], md1, true, m1, m8, m4); afs_rename# (; old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode0 ∧ new_parent_inode = new_parent_inode0 ∧ old_child_inode = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent1 = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ afs_dirs = dirs ∧ afs_files = (files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced))), str0 ∈ entries1, asynced1 ∧ ¬ asynced0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 60:
# links(entries[str], cache_dirs) = cache_dirs[entries[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, entries[str] = cache_dirs[m3].entries[str], cache_dirs[m3] = mkdir(cache_dirs[m3].size, md2, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries), cache_dirs[m] = mkdir(cache_dirs[m].size, md4, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), cache_dirs[entries[str]] = mkdir(cache_dirs[entries[str]].size, md5, cache_dirs[entries[str]].nlink, cache_dirs[entries[str]].nsubdirs, entries2), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs [m3, ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= (cache_dirs[m3].size - 1)).nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ] [m, ( ( (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= (cache_dirs[m3].size - 1)) .nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;cache_dirs[m] ) .entries:= (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= (cache_dirs[m3].size - 1)) .nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;cache_dirs[m] ) .entries [str0, entries[str] ]) .size:= ( (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= (cache_dirs[m3].size - 1)) .nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;cache_dirs[m] ) .size + 1)) .nsubdirs:= ( (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= (cache_dirs[m3].size - 1)) .nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;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), str ∈ cache_dirs[m3].entries, m ∈ cache_dirs, m3 ∈ cache_dirs, str ∈ entries, entries[str] ∈ cache_dirs, 1 ≤ cache_dirs[m3].size, 1 ≤ cache_dirs[m3].nsubdirs, ¬ str0 ∈ cache_dirs[m].entries, ¬ str0 ∈ entries1, 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, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size); 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), cache_dirs[m3].nsubdirs, cache_dirs[m3].size); 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)))

Rule argument

Additional information

Used simplifier rules: ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ mkdentry(str, m) ≠ negdentry(str0); (s)% ⊦ ¬ negdentry(str).dentry?; (s)% ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (s)% ⊦ mkdentry(str, m).ino = m; (s)% ⊦ negdentry(str).name = str; (s)% ⊦ (ad.entries:= entries).nsubdirs = ad.nsubdirs; (s)% ⊦ (ad.size:= m).nsubdirs = ad.nsubdirs; (s)% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)% ⊦ (ad.entries:= entries).size = ad.size; (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)% ⊦ mkdentry(str, m).name = str; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)