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Proof step 57 from proof refine-unlink (previous step)

# links(entries0[str], cache_dirs) = m8, # links(m, cache_dirs) = m9, cache_files[entries0[str]] = mkfile(m0, md4, # links(entries0[str], cache_dirs), content1), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m7, md2, # links(entries0[str], cache_dirs), content0), cache_dirs[m] = mkdir(m1, md1, # links(m, cache_dirs), m3, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6), dent = mkdentry(str, entries0[str]), dirs = dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)] [p_ino, dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino] .size:= (dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino].size - 1) ], dirs0 = cache_dirs, err = err0, files = cache_files, ino = dent.ino, p_ino = p_inode.ino, p_inode = mkinode(m, md, true, # links(m, cache_dirs), m3, m1), 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-unlink(p_inode, c_inode, dent, dirs0, files), synced(cache_dirs, cache_icache, cache_pcache), entries0[str] ∈ cache_files, str ∈ entries0, m ∈ cache_dirs, dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino].size ≥ 1, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(ino; dirs, files); dent := negdentry(dent.name); afs_inode_sync#(p_inode; dirs, files); afs_inode_sync#(c_inode; dirs, files); p_inode := getinode(p_inode.ino, dirs, files); c_inode := getinode(c_inode.ino, dirs, files); err := ESUCCESS; asynced := ?; if err = ESUCCESS then let asynced = false in {icache_set#(c_inode, asynced; err, cache_icache); icache_set#(p_inode, asynced; err, cache_icache); dcache_delete#(p_inode.ino, dent.name; err, cache_dcache)}] ( icache-cons(cache_icache, dirs, files) ∧ dcache-cons(cache_dcache, dirs, files) ∧ pcache-cons(cache_pcache, dirs, files) ∧ synced(dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(files, cache_icache, cache_pcache)) ∧ afs-cons(dirs, files) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m, md, true, m9, m3, m1); p_inode1 := mkinode(entries0[str], md0, false, m8, 0, m6); afs_unlink#(; p_inode0, p_inode1, dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = dirs ∧ afs_files = (files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 58:
# links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries0[str], cache_dirs) = cache_files[entries0[str]].nlink, entries0[str] = cache_dirs[m].entries[str], cache_files[entries0[str]] = mkfile(m0, md4, cache_files[entries0[str]].nlink, content1), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m7, md2, cache_files[entries0[str]].nlink, content0), cache_dirs[m] = mkdir(cache_dirs[m].size, md1, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, (cache_dirs[m].entries:= (cache_dirs[m].entries -- str)).size:= (cache_dirs[m].size - 1)], cache_dirs[m].size ≠ 0, 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[m].entries, entries0[str] ∈ cache_files, str ∈ entries0, m ∈ cache_dirs, 1 ≤ cache_dirs[m].size, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries0[str]).ino; dirs, cache_files); dent := negdentry(mkdentry(str, entries0[str]).name); afs_inode_sync#(mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, cache_files); afs_inode_sync#(mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); 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(entries0[str], dirs, cache_files); err := ESUCCESS; asynced := ?; if err = ESUCCESS then let asynced = false in {icache_set#(c_inode, asynced; err, cache_icache); icache_set#(p_inode, asynced; err, cache_icache); dcache_delete#(p_inode.ino, dent.name; err, cache_dcache)}] ( icache-cons(cache_icache, dirs, cache_files) ∧ dcache-cons(cache_dcache, dirs, cache_files) ∧ pcache-cons(cache_pcache, dirs, cache_files) ∧ synced(dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) ∧ afs-cons(dirs, cache_files) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); p_inode1 := mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); afs_unlink#(; 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))

Rule argument

Additional information

Used simplifier rules: afs-cons-file-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ files ∧ files[ino] = mkfile(m, md, n, content) → # links(ino, dirs) = n; (forward) from specification afs-base%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat%⊦ 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)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).size = m2; (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)% ⊦ 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)% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ mkdentry(str, m).ino = m; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (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)%⊦ pre-unlink(p_inode, c_inode, dent, dirs, files) ↔ valid-dentry(p_inode.ino, dent, dirs, files) ∧ valid-file-inode(c_inode, dirs, files) ∧ valid-dir-inode(p_inode, dirs, files) ∧ c_inode.ino = dent.ino; (s)% ⊦ n ≥ m ↔ m ≤ n; (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)