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Proof step 63 from proof refine-rename-dir-dentry (previous step)

# links(entries[str0], cache_dirs) = cache_dirs[entries[str0]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries1[str], cache_dirs) = cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].entries = ∅, cache_dirs[m3].nsubdirs = m1 + 1, cache_dirs[m3].size = m0 + 1, entries[str0] = cache_dirs[m3].entries[str0], entries1[str] = cache_dirs[m].entries[str], cache_dirs[m3] = mkdir(m0 + 1, md4, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries), cache_dirs[m] = mkdir(cache_dirs[m].size, md7, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), cache_dirs[entries1[str]] = mkdir(cache_dirs[entries1[str]].size, md6, cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].nsubdirs, ∅), cache_dirs[entries[str0]] = mkdir(cache_dirs[entries[str0]].size, md3, cache_dirs[entries[str0]].nlink, cache_dirs[entries[str0]].nsubdirs, entries2), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= m1] [m, (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= m1;cache_dirs[m]) .entries:= (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= m1;cache_dirs[m]).entries [str, entries[str0] ] ], 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), str0 ∈ cache_dirs[m3].entries, str ∈ cache_dirs[m].entries, m3 ∈ cache_dirs, m ∈ cache_dirs, entries[str0] ∈ cache_dirs, entries1[str] ∈ cache_dirs, str0 ∈ entries, str ∈ entries1, 1 ≤ cache_dirs[m3].nsubdirs, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries1[str]).ino; dirs, cache_files); old_dent := mkdentry(str0, entries[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries1[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, 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 := mkinode(entries1[str], md1, true, # links(entries1[str], cache_dirs), cache_dirs[entries1[str]].nsubdirs, cache_dirs[entries1[str]].size); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, cache_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, cache_files[new_child_inode.ino].meta, false, cache_files[new_child_inode.ino].nlink, 0, cache_files[new_child_inode.ino].size) )}; 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 := ?; 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); if mkdentry(str, entries1[str]).dentry? then icache_set#(new_child_inode, asynced; err, cache_icache); dcache_delete#(old_parent_inode.ino, old_dent.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)) → ⟨new_dent0 := mkdentry(str, entries1[str]); old_dent0 := mkdentry(str0, entries[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, m0 + 1); new_child_inode0 := mkinode (entries1[str], md1, true, # links(entries1[str], cache_dirs), cache_dirs[entries1[str]].nsubdirs, cache_dirs[entries1[str]].size); old_child_inode0 := mkinode (entries[str0], md2, true, # links(entries[str0], cache_dirs), cache_dirs[entries[str0]].nsubdirs, cache_dirs[entries[str0]].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[str0], md2, true, # links(entries[str0], cache_dirs0), cache_dirs1[entries[str0]].nsubdirs, cache_dirs2[entries[str0]].size) = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced))), cache_dirs[m3].nsubdirs = 0, entries1[str] = ROOT_INO, m3 = m ∧ mkdentry(str, entries1[str]) = mkdentry(str0, entries[str0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 64:
# links(entries[str0], cache_dirs) = cache_dirs[entries[str0]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries1[str], cache_dirs) = cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].entries = ∅, cache_dirs[m3].nsubdirs = m1 + 1, cache_dirs[m3].size = m0 + 1, entries[str0] = cache_dirs[m3].entries[str0], entries1[str] = cache_dirs[m].entries[str], cache_dirs[m3] = mkdir(m0 + 1, md4, cache_dirs[m3].nlink, m1 + 1, entries), cache_dirs[m] = mkdir(cache_dirs[m].size, md7, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), cache_dirs[entries1[str]] = mkdir(cache_dirs[entries1[str]].size, md6, cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].nsubdirs, ∅), cache_dirs[entries[str0]] = mkdir(cache_dirs[entries[str0]].size, md3, cache_dirs[entries[str0]].nlink, cache_dirs[entries[str0]].nsubdirs, entries2), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= m1] [m, (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= m1;cache_dirs[m]) .entries:= (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= m1;cache_dirs[m]).entries [str, entries[str0] ] ], entries1[str] ≠ ROOT_INO, 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), str0 ∈ cache_dirs[m3].entries, str ∈ cache_dirs[m].entries, m3 ∈ cache_dirs, m ∈ cache_dirs, entries[str0] ∈ cache_dirs, entries1[str] ∈ cache_dirs, str0 ∈ entries, str ∈ entries1, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries1[str]).ino; dirs, cache_files); old_dent := mkdentry(str0, entries[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries1[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m3, md0, true, cache_dirs[m3].nlink, 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 := mkinode(entries1[str], md1, true, # links(entries1[str], cache_dirs), cache_dirs[entries1[str]].nsubdirs, cache_dirs[entries1[str]].size); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, cache_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, cache_files[new_child_inode.ino].meta, false, cache_files[new_child_inode.ino].nlink, 0, cache_files[new_child_inode.ino].size) )}; 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 := ?; 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); if mkdentry(str, entries1[str]).dentry? then icache_set#(new_child_inode, asynced; err, cache_icache); dcache_delete#(old_parent_inode.ino, old_dent.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)) → ⟨new_dent0 := mkdentry(str, entries1[str]); old_dent0 := mkdentry(str0, entries[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); new_child_inode0 := mkinode (entries1[str], md1, true, # links(entries1[str], cache_dirs), cache_dirs[entries1[str]].nsubdirs, cache_dirs[entries1[str]].size); old_child_inode0 := mkinode (entries[str0], md2, true, # links(entries[str0], cache_dirs), cache_dirs[entries[str0]].nsubdirs, cache_dirs[entries[str0]].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[str0], md2, true, # links(entries[str0], cache_dirs0), cache_dirs1[entries[str0]].nsubdirs, cache_dirs2[entries[str0]].size) = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced))), m3 = m ∧ mkdentry(str, entries1[str]) = mkdentry(str0, entries[str0])

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ m + n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat