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Proof step 41 from proof refine-rmdir (previous step)

# links(entries[str], cache_dirs) = cache_dirs[entries[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[entries[str]].entries = ∅, cache_dirs[m].nsubdirs = m1 + 1, cache_dirs[m].size = m0 + 1, entries[str] = cache_dirs[m].entries[str], cache_dirs[entries[str]] = mkdir(cache_dirs[entries[str]].size, md2, cache_dirs[entries[str]].nlink, cache_dirs[entries[str]].nsubdirs, ∅), cache_dirs[m] = mkdir(m0 + 1, md3, cache_dirs[m].nlink, m1 + 1, entries), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, ((cache_dirs[m].entries:= (cache_dirs[m].entries -- str)).size:= m0).nsubdirs:= m1], entries[str] ≠ ROOT_INO, 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 ∈ cache_dirs[m].entries, m ∈ cache_dirs, str ∈ entries, entries[str] ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, cache_files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(m, md, true, cache_dirs[m].nlink, m1 + 1, m0 + 1); dirs, cache_files); afs_inode_sync# (mkinode(entries[str], md0, true, # links(entries[str], cache_dirs), cache_dirs[entries[str]].nsubdirs, cache_dirs[entries[str]].size) ; 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(entries[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, entries[str]); p_inode0 := mkinode(m, md, true, # links(m, cache_dirs), m1 + 1, m0 + 1); p_inode1 := mkinode(entries[str], md0, true, # links(entries[str], cache_dirs), cache_dirs[entries[str]].nsubdirs, cache_dirs[entries[str]].size); afs_rmdir#(; 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))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 42:
# links(m, cache_dirs) = m3, # links(entries[str], cache_dirs) = cache_dirs[entries[str]].nlink, cache_dirs[entries[str]].entries = ∅, cache_dirs[m] = mkdir(m0 + 1, md3, m3, m1 + 1, entries), cache_dirs[entries[str]] = mkdir(cache_dirs[entries[str]].size, md2, cache_dirs[entries[str]].nlink, cache_dirs[entries[str]].nsubdirs, ∅), entries[str] = entries0[str], cache_dirs[m] = mkdir(m2, md1, m3, m4, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, ((cache_dirs[m].entries:= (entries0 -- str)).size:= m0).nsubdirs:= m1], m2 = m0 + 1, m4 = m1 + 1, 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), entries[str] ∈ cache_dirs, str ∈ entries, m ∈ cache_dirs, str ∈ entries0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, cache_files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(m, md, true, m3, m1 + 1, m0 + 1); dirs, cache_files); afs_inode_sync# (mkinode(entries[str], md0, true, # links(entries[str], cache_dirs), cache_dirs[entries[str]].nsubdirs, cache_dirs[entries[str]].size) ; 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(entries[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, entries[str]); p_inode0 := mkinode(m, md, true, # links(m, cache_dirs), m1 + 1, m0 + 1); p_inode1 := mkinode(entries[str], md0, true, # links(entries[str], cache_dirs), cache_dirs[entries[str]].nsubdirs, cache_dirs[entries[str]].size); afs_rmdir#(; 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)), entries[str] = ROOT_INO

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [cache_dirs[m], m2, md1, m3, m4, entries0]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: cache_dirs[m], m2, md1, m3, m4, entries0