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Proof step 710 from proof refine-rename-dir-dentry (previous step)

# links(entries2[str], cache_dirs) = cache_dirs[entries2[str]].nlink, cache_dirs[entries2[str]].entries = ∅, cache_dirs[entries[str0]] = mkdir(m2, md6, # links(entries[str0], cache_dirs), m5, entries1), cache_dirs[entries2[str]] = mkdir(cache_dirs[entries2[str]].size, md4, cache_dirs[entries2[str]].nlink, cache_dirs[entries2[str]].nsubdirs, ∅), cache_dirs[m] = mkdir(m4, md3, # links(m, cache_dirs), m7, entries2), cache_dirs[m3] = mkdir(m0 + 1, md2, # links(m3, cache_dirs), m1 + 1, entries), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, mkdir(m0, md2, # links(m3, cache_dirs), m1, entries -- str0)] [m, (m = m3 ⊃ mkdir(m0, md2, # links(m3, cache_dirs), m1, entries -- str0);mkdir(m4, md3, # links(m, cache_dirs), m7, entries2)) .entries:= (m = m3 ⊃ mkdir(m0, md2, # links(m3, cache_dirs), m1, entries -- str0);mkdir(m4, md3, # links(m, cache_dirs), m7, entries2)) .entries [str, entries[str0] ] ], entries2[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 ∈ entries2, entries2[str] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, str0 ∈ entries, entries[str0] ∈ cache_dirs, ⌊err0⌋ , 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 ⊦ ⟨afs_drop_nlink#(mkdentry(str, entries2[str]).ino; dirs, afs_files); old_dent := mkdentry(str0, entries[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries2[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m3, md0, true, # links(m3, cache_dirs), m1 + 1, m0 + 1); dirs, afs_files); afs_inode_sync#(mkinode(m, md, true, # links(m, cache_dirs), m7, m4); dirs, afs_files); new_child_inode := mkinode(entries2[str], md1, true, # links(entries2[str], cache_dirs), cache_dirs[entries2[str]].nsubdirs, cache_dirs[entries2[str]].size); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, afs_files); new_child_inode := (new_child_inode.ino ∈ dirs ⊃ mkinode (new_child_inode.ino, dirs[new_child_inode.ino].meta, true, dirs[new_child_inode.ino].nlink, dirs[new_child_inode.ino].nsubdirs, dirs[new_child_inode.ino].size) ;mkinode (new_child_inode.ino, afs_files[new_child_inode.ino].meta, false, afs_files[new_child_inode.ino].nlink, 0, afs_files[new_child_inode.ino].size) )}; old_parent_inode := (m3 ∈ dirs ⊃ mkinode(m3, dirs[m3].meta, true, dirs[m3].nlink, dirs[m3].nsubdirs, dirs[m3].size) ;mkinode(m3, afs_files[m3].meta, false, afs_files[m3].nlink, 0, afs_files[m3].size) ); new_parent_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, afs_files[m].meta, false, afs_files[m].nlink, 0, afs_files[m].size) ); err := ESUCCESS; asynced := ?⟩ ( mkinode(m3, md0, true, # links(m3, cache_dirs), m1 + 1, m0 + 1) = old_parent_inode ∧ mkinode(m, md, true, # links(m, cache_dirs), m7, m4) = new_parent_inode ∧ mkinode (entries2[str], md1, true, # links(entries2[str], cache_dirs), cache_dirs[entries2[str]].nsubdirs, cache_dirs[entries2[str]].size) = new_child_inode ∧ mkdentry(str0, entries[str0]) = old_dent ∧ mkdentry(str, entries2[str]) = new_dent ∧ err0 = err ∧ dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (asynced ↔ asynced2)), ⟨fail#(; ; err); asynced := ?⟩ (err0 = err ∧ (asynced ↔ asynced2)), m3 = m ∧ mkdentry(str, entries2[str]) = mkdentry(str0, entries[str0])

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 711:
# links(entries2[str], cache_dirs) = m8, cache_dirs[m3] = mkdir(m0 + 1, md2, # links(m3, cache_dirs), m1 + 1, entries), cache_dirs[m] = mkdir(m4, md3, # links(m, cache_dirs), m7, entries2), cache_dirs[entries2[str]] = mkdir(m6, md4, m8, m9, ∅), cache_dirs[entries[str0]] = mkdir(m2, md6, # links(entries[str0], cache_dirs), m5, entries1), cache_dirs[entries2[str]] = mkdir(m6, md5, m8, m9, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, mkdir(m0, md2, # links(m3, cache_dirs), m1, entries -- str0)] [m, (m = m3 ⊃ mkdir(m0, md2, # links(m3, cache_dirs), m1, entries -- str0);mkdir(m4, md3, # links(m, cache_dirs), m7, entries2)) .entries:= (m = m3 ⊃ mkdir(m0, md2, # links(m3, cache_dirs), m1, entries -- str0);mkdir(m4, md3, # links(m, cache_dirs), m7, entries2)) .entries [str, entries[str0] ] ], entries0 = ∅, 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[str0] ∈ cache_dirs, str0 ∈ entries, m3 ∈ cache_dirs, m ∈ cache_dirs, entries2[str] ∈ cache_dirs, str ∈ entries2, ⌊err0⌋ , 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 ⊦ ⟨afs_drop_nlink#(mkdentry(str, entries2[str]).ino; dirs, afs_files); old_dent := mkdentry(str0, entries[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries2[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m3, md0, true, # links(m3, cache_dirs), m1 + 1, m0 + 1); dirs, afs_files); afs_inode_sync#(mkinode(m, md, true, # links(m, cache_dirs), m7, m4); dirs, afs_files); new_child_inode := mkinode(entries2[str], md1, true, # links(entries2[str], cache_dirs), m9, m6); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, afs_files); new_child_inode := (new_child_inode.ino ∈ dirs ⊃ mkinode (new_child_inode.ino, dirs[new_child_inode.ino].meta, true, dirs[new_child_inode.ino].nlink, dirs[new_child_inode.ino].nsubdirs, dirs[new_child_inode.ino].size) ;mkinode (new_child_inode.ino, afs_files[new_child_inode.ino].meta, false, afs_files[new_child_inode.ino].nlink, 0, afs_files[new_child_inode.ino].size) )}; old_parent_inode := (m3 ∈ dirs ⊃ mkinode(m3, dirs[m3].meta, true, dirs[m3].nlink, dirs[m3].nsubdirs, dirs[m3].size) ;mkinode(m3, afs_files[m3].meta, false, afs_files[m3].nlink, 0, afs_files[m3].size) ); new_parent_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, afs_files[m].meta, false, afs_files[m].nlink, 0, afs_files[m].size) ); err := ESUCCESS; asynced := ?⟩ ( mkinode(m3, md0, true, # links(m3, cache_dirs), m1 + 1, m0 + 1) = old_parent_inode ∧ mkinode(m, md, true, # links(m, cache_dirs), m7, m4) = new_parent_inode ∧ mkinode(entries2[str], md1, true, # links(entries2[str], cache_dirs), m9, m6) = new_child_inode ∧ mkdentry(str0, entries[str0]) = old_dent ∧ mkdentry(str, entries2[str]) = new_dent ∧ err0 = err ∧ dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (asynced ↔ asynced2)), ⟨fail#(; ; err); asynced := ?⟩ (err0 = err ∧ (asynced ↔ asynced2)), entries2[str] = ROOT_INO, m3 = m ∧ mkdentry(str, entries2[str]) = mkdentry(str0, entries[str0])

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[entries2[str]], m6, md5, m8, m9, 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[entries2[str]], m6, md5, m8, m9, entries0