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Proof step 549 from proof refine-rename-file-dentry (previous step)

((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str0]] = mkfile(m8, md4, # links(entries[str0], cache_dirs), content0), cache_files[entries[str0]] = mkfile(m1, md, # links(entries[str0], cache_dirs), content), cache_dirs[m2] = mkdir(m6, md3, # links(m2, cache_dirs), m5, entries), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, cache_dirs0 = cache_dirs, dirs = cache_dirs, err = err0, files = afs_files, new_child_inode = new_child_inode0, new_child_inode0 = mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs), 0, m), new_dent = new_dent0, new_dent0 = mkdentry(str0, entries[str0]), new_parent_inode = new_parent_inode0, new_parent_inode0 = mkinode(m2, md5, true, # links(m2, cache_dirs), m5, m6), old_child_inode = old_child_inode0, old_child_inode0 = mkinode(entries[str0], md2, false, # links(entries[str0], cache_dirs), 0, m13), old_dent = old_dent0, old_dent0 = mkdentry(str0, entries[str0]), old_parent_inode = old_parent_inode0, old_parent_inode0 = mkinode(m2, md5, true, # links(m2, cache_dirs), m5, m6), afs-cons (cache_dirs[m2, mkdir(m6, md5, # links(m2, cache_dirs), m5, entries)], cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)]), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), dcache-cons(cache_dcache, cache_dirs, cache_files), dcache-cons (cache_dcache[dentrykey(m2, str0), mkdentry(str0, entries[str0])], cache_dirs[m2, mkdir(m6, md5, # links(m2, cache_dirs), m5, entries)], cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)]), icache-cons(cache_icache, cache_dirs, cache_files), icache-cons (cache_icache[m2, mkientry(false, mkinode(m2, md5, true, # links(m2, cache_dirs), m5, m6))] [entries[str0], mkientry(false, mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs), 0, m)) ], cache_dirs[m2, mkdir(m6, md5, # links(m2, cache_dirs), m5, entries)], cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)]), pcache-cons(cache_pcache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs[m2, mkdir(m6, md5, # links(m2, cache_dirs), m5, entries)], cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)]), pre-rename(old_parent_inode, old_dent, old_child_inode, new_parent_inode, new_dent, new_child_inode, dirs, files), synced(cache_dirs, cache_icache, cache_pcache), synced (cache_dirs[m2, mkdir(m6, md5, # links(m2, cache_dirs), m5, entries)], cache_icache[m2, mkientry(false, mkinode(m2, md5, true, # links(m2, cache_dirs), m5, m6))] [entries[str0], mkientry(false, mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs), 0, m)) ], cache_pcache), str0 ∈ entries, entries[str0] ∈ cache_files, m2 ∈ cache_dirs, cache_sync → synced (cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)], cache_icache[m2, mkientry(false, mkinode(m2, md5, true, # links(m2, cache_dirs), m5, m6))] [entries[str0], mkientry(false, mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs), 0, m)) ], cache_pcache), cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m2, cache_dirs) ≠ ∅ ∨ m2 = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ ( (m2 ∈ cache_dirs0 ⊃ mkinode(m2, md5, true, # links(m2, cache_dirs0), m5, m6) ;mkinode (m2, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).meta, false, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).nlink, 0, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).size) ) = old_parent_inode ∧ (m2 ∈ cache_dirs0 ⊃ mkinode(m2, md5, true, # links(m2, cache_dirs0), m5, m6) ;mkinode (m2, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).meta, false, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).nlink, 0, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).size) ) = new_parent_inode ∧ mkinode(entries[str0], md2, false, # links(entries[str0], cache_dirs0), 0, m13) = old_child_inode ∧ (entries[str0] ∈ cache_dirs0 ⊃ mkinode (entries[str0], (entries[str0] = m2 ⊃ mkdir(m6, md5, # links(entries[str0], cache_dirs0), m5, entries);cache_dirs0[entries[str0]]).meta, true, (entries[str0] = m2 ⊃ mkdir(m6, md5, # links(entries[str0], cache_dirs0), m5, entries);cache_dirs0[entries[str0]]).nlink, (entries[str0] = m2 ⊃ mkdir(m6, md5, # links(entries[str0], cache_dirs0), m5, entries);cache_dirs0[entries[str0]]).nsubdirs, (entries[str0] = m2 ⊃ mkdir(m6, md5, # links(entries[str0], cache_dirs0), m5, entries);cache_dirs0[entries[str0]]).size) ;mkinode(entries[str0], md0, false, cache_files[entries[str0]].nlink, 0, m) ) = new_child_inode ∧ negdentry(str0) = old_dent ∧ mkdentry(str0, entries[str0]) = new_dent ∧ ESUCCESS = err ∧ dirs = cache_dirs0[m2, mkdir(m6, md5, # links(m2, cache_dirs0), m5, entries)] ∧ files = (cache_files[entries[str0], (cache_files[entries[str0]].size:= m).meta:= md0] ⊕ cache_pcache) ⊕ cache_icache[m2, mkientry(false, mkinode(m2, md5, true, # links(m2, cache_dirs0), m5, m6))] [entries[str0], mkientry(false, mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs0), 0, m)) ] ∧ (asynced ↔ asynced0)), ⟨let old_ino = old_parent_inode.ino, new_ino = new_parent_inode.ino, asynced1 = new_dent.dentry? in let asynced0 = old_ino = new_ino ∧ new_dent = old_dent in {dirs[old_ino].entries := dirs[old_ino].entries -- old_dent.name; if ¬ asynced0 then {dirs[old_ino].size := dirs[old_ino].size - 1; if old_dent.ino ∈ dirs then dirs[old_ino].nsubdirs := dirs[old_ino].nsubdirs - 1}; dirs[new_ino].entries[new_dent.name] := old_dent.ino; if asynced1 ∧ ¬ asynced0 then let ino = new_dent.ino in afs_drop_nlink#(ino; dirs, files); if ¬ asynced1 then {dirs[new_ino].size := dirs[new_ino].size + 1; if old_dent.ino ∈ dirs then dirs[new_ino].nsubdirs := dirs[new_ino].nsubdirs + 1}; new_dent := mkdentry(new_dent.name, old_dent.ino); old_dent := negdentry(old_dent.name); 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 := ?⟩ ( (m2 ∈ cache_dirs0 ⊃ mkinode(m2, md5, true, # links(m2, cache_dirs0), m5, m6) ;mkinode (m2, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).meta, false, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).nlink, 0, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).size) ) = old_parent_inode ∧ (m2 ∈ cache_dirs0 ⊃ mkinode(m2, md5, true, # links(m2, cache_dirs0), m5, m6) ;mkinode (m2, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).meta, false, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).nlink, 0, (m2 = entries[str0] ⊃ (cache_files[entries[str0]].size:= m).meta:= md0;cache_files[m2]).size) ) = new_parent_inode ∧ mkinode(entries[str0], md2, false, # links(entries[str0], cache_dirs0), 0, m13) = old_child_inode ∧ (entries[str0] ∈ cache_dirs0 ⊃ mkinode (entries[str0], (entries[str0] = m2 ⊃ mkdir(m6, md5, # links(entries[str0], cache_dirs0), m5, entries);cache_dirs0[entries[str0]]).meta, true, (entries[str0] = m2 ⊃ mkdir(m6, md5, # links(entries[str0], cache_dirs0), m5, entries);cache_dirs0[entries[str0]]).nlink, (entries[str0] = m2 ⊃ mkdir(m6, md5, # links(entries[str0], cache_dirs0), m5, entries);cache_dirs0[entries[str0]]).nsubdirs, (entries[str0] = m2 ⊃ mkdir(m6, md5, # links(entries[str0], cache_dirs0), m5, entries);cache_dirs0[entries[str0]]).size) ;mkinode(entries[str0], md0, false, cache_files[entries[str0]].nlink, 0, m) ) = new_child_inode ∧ negdentry(str0) = old_dent ∧ mkdentry(str0, entries[str0]) = new_dent ∧ ESUCCESS = err ∧ dirs = cache_dirs0[m2, mkdir(m6, md5, # links(m2, cache_dirs0), m5, entries)] ∧ files = (cache_files[entries[str0], (cache_files[entries[str0]].size:= m).meta:= md0] ⊕ cache_pcache) ⊕ cache_icache[m2, mkientry(false, mkinode(m2, md5, true, # links(m2, cache_dirs0), m5, m6))] [entries[str0], mkientry(false, mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs0), 0, m)) ] ∧ (asynced ↔ asynced0)), entries[str0] ∈ cache_dirs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 550:
# links(m2, cache_dirs) = cache_dirs[m2].nlink, entries[str0] = cache_dirs[m2].entries[str0], ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str0]] = mkfile(m8, md4, # links(entries[str0], cache_dirs), content0), cache_files[entries[str0]] = mkfile(m1, md, # links(entries[str0], cache_dirs), content), cache_dirs[m2] = mkdir(cache_dirs[m2].size, md3, cache_dirs[m2].nlink, cache_dirs[m2].nsubdirs, entries), afs-cons (cache_dirs[m2, mkdir(cache_dirs[m2].size, md5, cache_dirs[m2].nlink, cache_dirs[m2].nsubdirs, entries)], cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)]), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), dcache-cons(cache_dcache, cache_dirs, cache_files), dcache-cons (cache_dcache[dentrykey(m2, str0), mkdentry(str0, entries[str0])], cache_dirs[m2, mkdir(cache_dirs[m2].size, md5, cache_dirs[m2].nlink, cache_dirs[m2].nsubdirs, entries)], cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)]), icache-cons(cache_icache, cache_dirs, cache_files), icache-cons (cache_icache[m2, mkientry(false, mkinode(m2, md5, true, cache_dirs[m2].nlink, cache_dirs[m2].nsubdirs, cache_dirs[m2].size))] [entries[str0], mkientry(false, mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs), 0, m)) ], cache_dirs[m2, mkdir(cache_dirs[m2].size, md5, cache_dirs[m2].nlink, cache_dirs[m2].nsubdirs, entries)], cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)]), pcache-cons(cache_pcache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs[m2, mkdir(cache_dirs[m2].size, md5, cache_dirs[m2].nlink, cache_dirs[m2].nsubdirs, entries)], cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)]), synced(cache_dirs, cache_icache, cache_pcache), synced (cache_dirs[m2, mkdir(cache_dirs[m2].size, md5, cache_dirs[m2].nlink, cache_dirs[m2].nsubdirs, entries)], cache_icache[m2, mkientry(false, mkinode(m2, md5, true, cache_dirs[m2].nlink, cache_dirs[m2].nsubdirs, cache_dirs[m2].size))] [entries[str0], mkientry(false, mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs), 0, m)) ], cache_pcache), str0 ∈ cache_dirs[m2].entries, str0 ∈ entries, entries[str0] ∈ cache_files, m2 ∈ cache_dirs, ¬ entries[str0] ∈ cache_dirs, cache_sync → synced (cache_files[entries[str0], mkfile(m, md0, # links(entries[str0], cache_dirs), content)], cache_icache[m2, mkientry(false, mkinode(m2, md5, true, cache_dirs[m2].nlink, cache_dirs[m2].nsubdirs, cache_dirs[m2].size))] [entries[str0], mkientry(false, mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs), 0, m)) ], cache_pcache), cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m2, cache_dirs) ≠ ∅ ∨ m2 = ROOT_INO ⊦ ⟨let old_ino = m2, new_ino = m2, asynced1 = true in let asynced0 = old_ino = new_ino ∧ mkdentry(str0, entries[str0]) = mkdentry(str0, entries[str0]) in {dirs := cache_dirs[old_ino, cache_dirs[old_ino].entries:= (cache_dirs[old_ino].entries -- mkdentry(str0, entries[str0]).name)]; old_dent := mkdentry(str0, entries[str0]); if ¬ asynced0 then {dirs[old_ino].size := dirs[old_ino].size - 1; if old_dent.ino ∈ dirs then dirs[old_ino].nsubdirs := dirs[old_ino].nsubdirs - 1}; dirs[new_ino].entries[mkdentry(str0, entries[str0]).name] := old_dent.ino; new_dent := mkdentry(str0, entries[str0]); afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; if asynced1 ∧ ¬ asynced0 then let ino = new_dent.ino in afs_drop_nlink#(ino; dirs, afs_files); if ¬ asynced1 then {dirs[new_ino].size := dirs[new_ino].size + 1; if old_dent.ino ∈ dirs then dirs[new_ino].nsubdirs := dirs[new_ino].nsubdirs + 1}; new_dent := mkdentry(new_dent.name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m2, md5, true, # links(m2, cache_dirs), cache_dirs[m2].nsubdirs, cache_dirs[m2].size); dirs, afs_files); afs_inode_sync#(mkinode(m2, md5, true, # links(m2, cache_dirs), cache_dirs[m2].nsubdirs, cache_dirs[m2].size); dirs, afs_files); new_child_inode := mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs), 0, m); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, afs_files); new_child_inode := (new_child_inode.ino ∈ dirs ⊃ mkinode (new_child_inode.ino, dirs[new_child_inode.ino].meta, true, dirs[new_child_inode.ino].nlink, dirs[new_child_inode.ino].nsubdirs, dirs[new_child_inode.ino].size) ;mkinode (new_child_inode.ino, afs_files[new_child_inode.ino].meta, false, afs_files[new_child_inode.ino].nlink, 0, afs_files[new_child_inode.ino].size) )}; old_parent_inode := (m2 ∈ dirs ⊃ mkinode(m2, dirs[m2].meta, true, dirs[m2].nlink, dirs[m2].nsubdirs, dirs[m2].size) ;mkinode(m2, afs_files[m2].meta, false, afs_files[m2].nlink, 0, afs_files[m2].size) ); new_parent_inode := (m2 ∈ dirs ⊃ mkinode(m2, dirs[m2].meta, true, dirs[m2].nlink, dirs[m2].nsubdirs, dirs[m2].size) ;mkinode(m2, afs_files[m2].meta, false, afs_files[m2].nlink, 0, afs_files[m2].size) ); err := ESUCCESS}; asynced := ?⟩ ( mkinode(m2, md5, true, # links(m2, cache_dirs), cache_dirs[m2].nsubdirs, cache_dirs[m2].size) = old_parent_inode ∧ mkinode(m2, md5, true, # links(m2, cache_dirs), cache_dirs[m2].nsubdirs, cache_dirs[m2].size) = new_parent_inode ∧ mkinode(entries[str0], md0, false, cache_files[entries[str0]].nlink, 0, m) = new_child_inode ∧ negdentry(str0) = old_dent ∧ mkdentry(str0, entries[str0]) = new_dent ∧ ESUCCESS = err ∧ dirs = cache_dirs[m2, mkdir(cache_dirs[m2].size, md5, # links(m2, cache_dirs), cache_dirs[m2].nsubdirs, entries)] ∧ afs_files = (cache_files[entries[str0], (cache_files[entries[str0]].size:= m).meta:= md0] ⊕ cache_pcache) ⊕ cache_icache[m2, mkientry(false, mkinode(m2, md5, true, # links(m2, cache_dirs), cache_dirs[m2].nsubdirs, cache_dirs[m2].size))] [entries[str0], mkientry(false, mkinode(entries[str0], md0, false, # links(entries[str0], cache_dirs), 0, m)) ] ∧ (asynced ↔ asynced0))

Rule argument

Additional information

Used simplifier rules: ⊦ mkfile(m, md, m0, content).nlink = m0; (s)%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%⊦ getinode(ino, dirs, files) = (ino ∈ dirs ⊃ mkinode (ino, dirs[ino].meta, true, dirs[ino].nlink, dirs[ino].nsubdirs, dirs[ino].size) ;mkinode (ino, files[ino].meta, false, files[ino].nlink, 0, files[ino].size) ); (s)% ⊦ mkdentry(str, m) ≠ negdentry(str0); (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).size = m2; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).directory ↔ boolvar; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nsubdirs = m1; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nlink = m0; (s)% ⊦ valid-file-ino(ino, dirs, files) ↔ ino ∈ files; (s)%⊦ valid-file-inode(inode, dirs, files) ↔ valid-file-ino(inode.ino, dirs, files) ∧ inode.nlink = files[inode.ino].nlink ∧ inode.nsubdirs = 0 ∧ ¬ inode.directory; (s)% ⊦ valid-dir-ino(ino, dirs, files) ↔ ino ∈ dirs; (s)%⊦ valid-dir-inode(inode, dirs, files) ↔ valid-dir-ino(inode.ino, dirs, files) ∧ inode.nlink = dirs[inode.ino].nlink ∧ inode.size = dirs[inode.ino].size ∧ inode.nsubdirs = dirs[inode.ino].nsubdirs ∧ inode.directory; (s)%⊦ valid-wb-inode(inode, dirs, files) ↔ valid-dir-inode(inode, dirs, files) ∨ valid-file-inode(inode, dirs, files); (s)% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ ino ∈ files ⊕ pcache ↔ ino ∈ files; (s)% ⊦ ino ∈ files ⊕ icache ↔ ino ∈ files; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ mkdentry(str, m).ino = m; (s)%⊦ valid-dentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ valid-ino(dent.ino, dirs, files) ∧ dent.name ∈ dirs[ino].entries ∧ dent.ino = dirs[ino].entries[dent.name] ∧ dent.dentry?; (s)% ⊦ ¬ mkdentry(str, m).negdentry?; (s)% ⊦ mkdentry(str, m).name = str; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (s)%⊦ valid-negdentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ ¬ dent.name ∈ dirs[ino].entries ∧ dent.negdentry?; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)%⊦ pre-rename (old_parent_inode, old_dent, old_child_inode, new_parent_inode, new_dent, new_child_inode, dirs, files) ↔ ( valid-negdentry(new_parent_inode.ino, new_dent, dirs, files) ∨ valid-dentry(new_parent_inode.ino, new_dent, dirs, files) ∧ valid-wb-inode(new_child_inode, dirs, files) ∧ new_child_inode.ino = new_dent.ino ∧ ( valid-file-ino(new_dent.ino, dirs, files) ↔ valid-file-ino(old_dent.ino, dirs, files)) ∧ ( valid-dir-ino(new_dent.ino, dirs, files) → new_dent.ino ≠ ROOT_INO ∧ dir-empty(new_dent.ino, dirs, files))) ∧ valid-dentry(old_parent_inode.ino, old_dent, dirs, files) ∧ valid-dir-inode(old_parent_inode, dirs, files) ∧ valid-dir-inode(new_parent_inode, dirs, files) ∧ ( new_parent_inode.ino = old_parent_inode.ino → new_parent_inode.meta = old_parent_inode.meta) ∧ valid-wb-inode(old_child_inode, dirs, files) ∧ old_child_inode.ino = old_dent.ino; (s)