Proof step 87 from proof refine-rmdir (previous step)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 88:

  (entries0[str] = m ⊃ mkdir(m0, md1, # links(entries0[str], cache_dirs), m1, entries0 -- str) ;mkdir(m2, md3, # links(entries0[str], cache_dirs), m4, ∅) ) = mkdir(m5, md2, m3 + 1, m7, entries), cache_dirs[entries0[str]] = mkdir(m2, md3, # links(entries0[str], cache_dirs), m4, ∅), cache_dirs[m] = mkdir(m0 + 1, md1, # links(m, cache_dirs), m1 + 1, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = ESUCCESS, entries0[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 ∈ entries0, m ∈ cache_dirs, entries0[str] ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [icache_set# (getinode (entries0[str], cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= md0 ], cache_files), false ; err, cache_icache); icache_set# (( m ∈ cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] ⊃ mkinode (m, cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m ] .meta, true, cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m ] .nlink, cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m ] .nsubdirs, cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m ] .size) ;mkinode(m, cache_files[m].meta, false, cache_files[m].nlink, 0, cache_files[m].size) ), false ; err, cache_icache); dcache_delete# (( m ∈ cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] ⊃ mkinode (m, cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m ] .meta, true, cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m ] .nlink, cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m ] .nsubdirs, cache_dirs[m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m ] .size) ;mkinode(m, cache_files[m].meta, false, cache_files[m].nlink, 0, cache_files[m].size) ) .ino, mkdentry(str, entries0[str]).name ; err, cache_dcache)] ( icache-cons (cache_icache, cache_dirs [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) ∧ dcache-cons (cache_dcache, cache_dirs [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) ∧ pcache-cons (cache_pcache, cache_dirs [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) ∧ synced (cache_dirs [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) ∧ afs-cons (cache_dirs [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m, md, true, # links(m, cache_dirs), m1 + 1, m0 + 1); p_inode1 := mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2); cache_dirs0 := cache_dirs; cache_dirs1 := cache_dirs; cache_dirs2 := cache_dirs; cache_dirs3 := cache_dirs; afs_rmdir#(; p_inode0, p_inode1, dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = cache_dirs2 [m, (m = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m, cache_dirs2), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m ∧ entries0[str] ∈ cache_dirs2 ⊃ mkdir(m5, md, m3, m7, entries) ;(entries0[str] ∈ cache_dirs2 ⊃ mkdir(m5, md2, m3, m7, entries);cache_dirs2[entries0[str]]) ) .meta:= md0 ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced0) ∧ (m ∈ cache_dirs2 ⊃ mkinode (m, (m = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m, cache_dirs2), m1, entries0 -- str)).meta, true, (m = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m, cache_dirs2), m1, entries0 -- str)).nlink, (m = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m, cache_dirs2), m1, entries0 -- str)).nsubdirs, (m = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m, cache_dirs2), m1, entries0 -- str)).size) ;mkinode(m, cache_files[m].meta, false, cache_files[m].nlink, 0, cache_files[m].size) ) = p_inode0 ∧ (entries0[str] ∈ cache_dirs3 ⊃ mkinode (entries0[str], md0, true, (entries0[str] = m ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).nlink, (entries0[str] = m ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).nsubdirs, (entries0[str] = m ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).size) ;mkinode(entries0[str], cache_files[entries0[str]].meta, false, cache_files[entries0[str]].nlink, 0, cache_files[entries0[str]].size) ) = p_inode1 ∧ negdentry(str) = dent0 ∧ err = err0))

Rule argument

Additional information

Used simplifier rules: ⊦ mkdentry(str, m).name = str; (s)% ⊦ (ad.meta:= md).size = ad.size; (s)% ⊦ (ad.meta:= md).nsubdirs = ad.nsubdirs; (s)% ⊦ (ad.meta:= md).nlink = ad.nlink; (s)% ⊦ (ad.meta:= md).meta = md; (s)%⊦ getinode(ino, dirs, files) = (ino ∈ dirs ⊃ mkinode (ino, dirs[ino].meta, true, dirs[ino].nlink, dirs[ino].nsubdirs, dirs[ino].size) ;mkinode (ino, files[ino].meta, false, files[ino].nlink, 0, files[ino].size) ); (s)%⊦ mkdir(m, md, m0, m1, entries).meta:= md0 = mkdir(m, md0, m0, m1, entries); (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)% ⊦ negdentry(str).name = str; (s)% ⊦ dirs[ino, ad][ino0, ad0][ino, ad1] = dirs[ino0, ad0][ino, ad1]; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).meta = md; (s)%⊦ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) ∧ str ∈ entries ∧ afs-cons(dirs, files) → (entries[str] ∈ files ↔ ¬ entries[str] ∈ dirs); (ss)% ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)