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Proof step 374 from proof refine-rename-dir-negdentry (previous step)

# links(entries[str], cache_dirs) = cache_dirs[entries[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, entries[str] = cache_dirs[m3].entries[str], cache_dirs[m3] = mkdir(cache_dirs[m3].size, md2, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries), cache_dirs[m] = mkdir(cache_dirs[m].size, md4, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), cache_dirs[entries[str]] = mkdir(cache_dirs[entries[str]].size, md5, cache_dirs[entries[str]].nlink, cache_dirs[entries[str]].nsubdirs, entries2), 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[m3].entries, m ∈ cache_dirs, m3 ∈ cache_dirs, entries[str] ∈ cache_dirs, str ∈ entries, ¬ str0 ∈ cache_dirs[m].entries, ¬ str0 ∈ entries1, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ [fail#(; ; err); asynced := ?; old_parent_inode := mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size); old_dent := mkdentry(str, entries[str]); new_parent_inode := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); new_dent := negdentry(str0); 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); skip; 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)) → ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(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, cache_dirs[m3].size); old_child_inode0 := mkinode(entries[str], md1, true, # links(entries[str], cache_dirs), cache_dirs[entries[str]].nsubdirs, cache_dirs[entries[str]].size); new_child_inode1 := new_child_inode0; 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[str], md1, true, # links(entries[str], cache_dirs0), cache_dirs1[entries[str]].nsubdirs, cache_dirs2[entries[str]].size) = old_child_inode0 ∧ new_child_inode1 = 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)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 375:
# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = m1, # links(entries[str], cache_dirs) = cache_dirs[entries[str]].nlink, cache_dirs[entries[str]] = mkdir(cache_dirs[entries[str]].size, md5, cache_dirs[entries[str]].nlink, cache_dirs[entries[str]].nsubdirs, entries2), cache_dirs[m] = mkdir(m0, md4, m1, m2, entries1), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md2, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries), entries[str] = cache_dirs[m3].entries[str], cache_dirs[m] = mkdir(m0, md3, m1, m2, entries0), 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 ∈ entries, entries[str] ∈ cache_dirs, m3 ∈ cache_dirs, m ∈ cache_dirs, str ∈ cache_dirs[m3].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [fail#(; ; err); asynced := ?; old_parent_inode := mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size); old_dent := mkdentry(str, entries[str]); new_parent_inode := mkinode(m, md, true, # links(m, cache_dirs), m2, m0); new_dent := negdentry(str0); 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); skip; 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)) → ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(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), cache_dirs[m3].nsubdirs, cache_dirs[m3].size); old_child_inode0 := mkinode(entries[str], md1, true, # links(entries[str], cache_dirs), cache_dirs[entries[str]].nsubdirs, cache_dirs[entries[str]].size); new_child_inode1 := new_child_inode0; 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[str], md1, true, # links(entries[str], cache_dirs0), cache_dirs1[entries[str]].nsubdirs, cache_dirs2[entries[str]].size) = old_child_inode0 ∧ new_child_inode1 = 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))), str0 ∈ entries0, str0 ∈ entries1

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], m0, md3, m1, m2, 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], m0, md3, m1, m2, entries0