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Proof step 746 from proof refine-rename-file-dentry (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, entries1[str0] = cache_dirs[m3].entries[str0], entries0[str] = cache_dirs[m].entries[str], cache_files[entries0[str]] = mkfile(m1, md5, # links(entries0[str], cache_dirs), content2), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md9, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m12, md3, # links(entries0[str], cache_dirs), content), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries1[str0]] = mkfile(m8, md4, # links(entries1[str0], cache_dirs), content0), cache_dirs[m] = mkdir(cache_dirs[m].size, md8, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_files[entries1[str0]] = mkfile(m0, md6, # links(entries1[str0], cache_dirs), content3), 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), str0 ∈ cache_dirs[m3].entries, str ∈ cache_dirs[m].entries, entries1[str0] ∈ cache_files, entries0[str] ∈ cache_files, m3 ∈ cache_dirs, m ∈ cache_dirs, str0 ∈ entries1, str ∈ entries0, ¬ entries0[str] ∈ cache_dirs, 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, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size); old_dent := mkdentry(str0, entries1[str0]); new_parent_inode := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); new_dent := mkdentry(str, entries0[str]); new_child_inode := mkinode(entries0[str], md1, false, # links(entries0[str], cache_dirs), 0, m10); 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), 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); new_child_inode0 := mkinode(entries0[str], md1, false, # links(entries0[str], cache_dirs), 0, m10); old_child_inode0 := mkinode(entries1[str0], md2, false, # links(entries1[str0], cache_dirs), 0, m13); cache_dirs0 := 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, false, # links(entries1[str0], cache_dirs0), 0, m13) = 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)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 747:
# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = m4, cache_files[entries1[str0]] = mkfile(m0, md6, # links(entries1[str0], cache_dirs), content3), cache_dirs[m] = mkdir(m2, md8, m4, m5, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries1[str0]] = mkfile(m8, md4, # links(entries1[str0], cache_dirs), content0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m12, md3, # links(entries0[str], cache_dirs), content), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md9, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), cache_files[entries0[str]] = mkfile(m1, md5, # links(entries0[str], cache_dirs), content2), entries0[str] = entries[str], entries1[str0] = cache_dirs[m3].entries[str0], cache_dirs[m] = mkdir(m2, md7, m4, m5, entries), 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 ∈ entries1, m ∈ cache_dirs, m3 ∈ cache_dirs, entries0[str] ∈ cache_files, entries1[str0] ∈ cache_files, str ∈ entries, str0 ∈ cache_dirs[m3].entries, 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, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size); old_dent := mkdentry(str0, entries1[str0]); new_parent_inode := mkinode(m, md, true, # links(m, cache_dirs), m5, m2); new_dent := mkdentry(str, entries0[str]); new_child_inode := mkinode(entries0[str], md1, false, # links(entries0[str], cache_dirs), 0, m10); 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), m5, m2); old_parent_inode0 := mkinode(m3, md0, true, # links(m3, cache_dirs), cache_dirs[m3].nsubdirs, cache_dirs[m3].size); new_child_inode0 := mkinode(entries0[str], md1, false, # links(entries0[str], cache_dirs), 0, m10); old_child_inode0 := mkinode(entries1[str0], md2, false, # links(entries1[str0], cache_dirs), 0, m13); cache_dirs0 := 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, false, # links(entries1[str0], cache_dirs0), 0, m13) = 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] ∈ cache_dirs

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, md7, m4, m5, entries]

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, md7, m4, m5, entries