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Proof step 22 from proof refine-rmdir (previous step)

# links(entries0[str], cache_dirs) = m8, # links(m, cache_dirs) = m5, asynced0 ↔ cache_asynced, cache_dirs[m] = mkdir(m1, md1, # links(m, cache_dirs), m7, entries0), cache_dirs[entries0[str]] = mkdir(m0, md3, # links(entries0[str], cache_dirs), m2, ∅), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m2, m0), dcache = cache_dcache, dent = mkdentry(str, entries0[str]), dirs = cache_dirs, err = err0, files = cache_files, icache = cache_icache, p_inode = mkinode(m, md, true, # links(m, cache_dirs), m7, m1), 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-rmdir(p_inode, c_inode, dent, dirs, files), synced(cache_dirs, cache_icache, cache_pcache), entries0[str] ∈ cache_dirs, str ∈ entries0, m ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_rmdir#(; p_inode, c_inode, dent, err, dirs, files; asynced0); if err = ESUCCESS then let asynced = false in {icache_set#(c_inode, asynced; err, icache); icache_set#(p_inode, asynced; err, icache); dcache_delete#(p_inode.ino, dent.name; err, dcache)}] ( icache-cons(icache, dirs, files) ∧ dcache-cons(dcache, dirs, files) ∧ pcache-cons(cache_pcache, dirs, files) ∧ synced(dirs, icache, cache_pcache) ∧ (cache_sync → synced(files, icache, cache_pcache)) ∧ afs-cons(dirs, files) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m, md, true, m5, m7, m1); p_inode1 := mkinode(entries0[str], md0, true, m8, m2, m0); afs_rmdir#(; p_inode0, p_inode1, dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = dirs ∧ afs_files = (files ⊕ cache_pcache) ⊕ icache ∧ (afs_asynced ↔ asynced0) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0)), entries0[str] = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 23:
# links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[entries0[str]].entries = ∅, cache_dirs[entries0[str]].nlink = m8, cache_dirs[m].nlink = m5, entries0[str] = cache_dirs[m].entries[str], cache_dirs[m] = mkdir(cache_dirs[m].size, md1, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md3, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(entries0[str], md0, true, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size), dent = mkdentry(str, entries0[str]), err = err0, m0 = cache_dirs[entries0[str]].size, m1 = cache_dirs[m].size, m2 = cache_dirs[entries0[str]].nsubdirs, m7 = cache_dirs[m].nsubdirs, p_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), entries0[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), str ∈ cache_dirs[m].entries, entries0[str] ∈ cache_dirs, str ∈ entries0, m ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_rmdir#(; p_inode, c_inode, dent, err, cache_dirs, cache_files; asynced0); 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, cache_dirs, cache_files) ∧ dcache-cons(cache_dcache, cache_dirs, cache_files) ∧ pcache-cons(cache_pcache, cache_dirs, cache_files) ∧ synced(cache_dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) ∧ afs-cons(cache_dirs, cache_files) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m, md, true, m5, m7, m1); p_inode1 := mkinode(entries0[str], md0, true, m8, m2, m0); afs_rmdir#(; p_inode0, p_inode1, dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced0) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0))

Rule argument

Additional information

Used simplifier rules: 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% ⊦ 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).size = m2; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nlink = m0; (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)% ⊦ dir-empty(ino, dirs, files) ↔ dirs[ino].entries = ∅; (s)% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ mkdentry(str, m).name = str; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (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)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)% ⊦ mkdentry(str, m).ino = m; (s)%⊦ pre-rmdir(p_inode, c_inode, dent, dirs, files) ↔ dent.ino ≠ ROOT_INO ∧ valid-dentry(p_inode.ino, dent, dirs, files) ∧ dir-empty(dent.ino, dirs, files) ∧ valid-dir-inode(p_inode, dirs, files) ∧ valid-dir-inode(c_inode, dirs, files) ∧ c_inode.ino = dent.ino; (s)