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Proof step 653 from proof refine-rename-dir-dentry (previous step)

# links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, # links(m3, cache_dirs) = m4, # links(entries[str0], cache_dirs) = cache_dirs[entries[str0]].nlink, cache_dirs[entries0[str]].entries = ∅, cache_dirs[entries[str0]] = mkdir(cache_dirs[entries[str0]].size, md3, cache_dirs[entries[str0]].nlink, cache_dirs[entries[str0]].nsubdirs, entries2), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md6, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), cache_dirs[m] = mkdir(m0, md5, # links(m, cache_dirs), m2, entries0), cache_dirs[m3] = mkdir(m1, md4, m4, m5, entries), entries[str0] = entries1[str0], cache_dirs[m3] = mkdir(m1, md7, m4, m5, entries1), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), 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 ∈ entries0, str0 ∈ entries, entries0[str] ∈ cache_dirs, entries[str0] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, str0 ∈ entries1, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ [fail#(; ; err); asynced := ?; old_parent_inode := mkinode(m3, md0, true, m4, m5, m1); old_dent := mkdentry(str0, entries[str0]); new_parent_inode := mkinode(m, md, true, # links(m, cache_dirs), m2, m0); new_dent := mkdentry(str, entries0[str]); new_child_inode := mkinode(entries0[str], md1, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size); afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; 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, entries0[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, cache_dirs, cache_files) ∧ dcache-cons(cache_dcache, cache_dirs, cache_files) ∧ synced(cache_dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨new_dent0 := mkdentry(str, entries0[str]); old_dent0 := mkdentry(str0, entries[str0]); new_parent_inode0 := mkinode(m, md, true, # links(m, cache_dirs), m2, m0); old_parent_inode0 := mkinode(m3, md0, true, # links(m3, cache_dirs), m5, m1); new_child_inode0 := mkinode (entries0[str], md1, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[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 = cache_dirs0 ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced))), entries0[str] = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 654:
# links(entries1[str0], cache_dirs) = cache_dirs[entries1[str0]].nlink, # links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].entries = ∅, cache_dirs[m3] = mkdir(m1, md7, # links(m3, cache_dirs), m5, entries1), cache_dirs[m] = mkdir(m0, md5, # links(m, cache_dirs), m2, entries0), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md6, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), cache_dirs[entries1[str0]] = mkdir(cache_dirs[entries1[str0]].size, md3, cache_dirs[entries1[str0]].nlink, cache_dirs[entries1[str0]].nsubdirs, entries2), 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), m3 ∈ cache_dirs, m ∈ cache_dirs, entries1[str0] ∈ cache_dirs, entries0[str] ∈ cache_dirs, str0 ∈ entries1, str ∈ entries0, 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 ⊦ [fail#(; ; err); asynced := ?; old_parent_inode := mkinode(m3, md0, true, # links(m3, cache_dirs), m5, m1); old_dent := mkdentry(str0, entries1[str0]); new_parent_inode := mkinode(m, md, true, # links(m, cache_dirs), m2, m0); new_dent := mkdentry(str, entries0[str]); new_child_inode := mkinode(entries0[str], md1, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size); afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; 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, entries0[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, cache_dirs, cache_files) ∧ dcache-cons(cache_dcache, cache_dirs, cache_files) ∧ synced(cache_dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨new_dent0 := mkdentry(str, entries0[str]); old_dent0 := mkdentry(str0, entries1[str0]); new_parent_inode0 := mkinode(m, md, true, # links(m, cache_dirs), m2, m0); old_parent_inode0 := mkinode(m3, md0, true, # links(m3, cache_dirs), m5, m1); new_child_inode0 := mkinode (entries0[str], md1, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size); old_child_inode0 := mkinode (entries1[str0], md2, true, # links(entries1[str0], cache_dirs), cache_dirs[entries1[str0]].nsubdirs, cache_dirs[entries1[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 (entries1[str0], md2, true, # links(entries1[str0], cache_dirs0), cache_dirs1[entries1[str0]].nsubdirs, cache_dirs2[entries1[str0]].size) = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ cache_dirs = cache_dirs0 ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced)))

Rule argument

Additional information

Used simplifier rules: ⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)%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