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Proof step 92 from proof refine-rmdir (previous step)

(entries0[str] = m6 ⊃ 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), asynced ↔ false, cache_dirs[m6] = mkdir(m0 + 1, md1, # links(m6, cache_dirs), m1 + 1, entries0), cache_dirs[entries0[str]] = mkdir(m2, md3, # links(entries0[str], cache_dirs), m4, ∅), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = ESUCCESS, icache = icache0[inode.ino, mkientry(asynced, inode)], icache0 = cache_icache, inode = getinode (entries0[str], cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= md0 ], cache_files), 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), entries0[str] ∈ cache_dirs, m6 ∈ cache_dirs, str ∈ entries0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ [icache_set# (( m6 ∈ cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] ⊃ mkinode (m6, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m6 ] .meta, true, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m6 ] .nlink, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m6 ] .nsubdirs, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), m4, m2).meta ] [m6 ] .size) ;mkinode(m6, cache_files[m6].meta, false, cache_files[m6].nlink, 0, cache_files[m6].size) ), false ; err, icache); dcache_delete#(m6, mkdentry(str, entries0[str]).name; err, cache_dcache)] ( icache-cons (icache, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) ∧ dcache-cons (cache_dcache, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) ∧ pcache-cons (cache_pcache, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) ∧ synced (cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], icache, cache_pcache) ∧ (cache_sync → synced(cache_files, icache, cache_pcache)) ∧ afs-cons (cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m6, md, true, # links(m6, 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 [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs2), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ∧ 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) ⊕ icache ∧ (afs_asynced ↔ asynced0) ∧ (m6 ∈ cache_dirs2 ⊃ mkinode (m6, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs2), m1, entries0 -- str)).meta, true, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs2), m1, entries0 -- str)).nlink, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs2), m1, entries0 -- str)).nsubdirs, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs2), m1, entries0 -- str)).size) ;mkinode(m6, cache_files[m6].meta, false, cache_files[m6].nlink, 0, cache_files[m6].size) ) = p_inode0 ∧ (entries0[str] ∈ cache_dirs3 ⊃ mkinode (entries0[str], md0, true, (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).nlink, (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).nsubdirs, (entries0[str] = m6 ⊃ 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)), entries0[str] = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 93:
(entries0[str] = m6 ⊃ 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[m6] = mkdir(m0 + 1, md1, # links(m6, cache_dirs), m1 + 1, entries0), err = ESUCCESS, icache = cache_icache [entries0[str], mkientry (false, mkinode (entries0[str], md0, true, (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).nlink, (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).nsubdirs, (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).size)) ], 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, m6 ∈ cache_dirs, entries0[str] ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ [icache_set# (mkinode (m6, cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= md0 ] [m6 ] .meta, true, cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= md0 ] [m6 ] .nlink, cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= md0 ] [m6 ] .nsubdirs, cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md] [entries0[str], cache_dirs[entries0[str], mkdir(m5, md2, m3, m7, entries)] [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str] ] .meta:= md0 ] [m6 ] .size), false ; err, icache); dcache_delete#(m6, mkdentry(str, entries0[str]).name; err, cache_dcache)] ( icache-cons (icache, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) ∧ dcache-cons (cache_dcache, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) ∧ pcache-cons (cache_pcache, cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) ∧ synced (cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], icache, cache_pcache) ∧ (cache_sync → synced(cache_files, icache, cache_pcache)) ∧ afs-cons (cache_dirs [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ], cache_files) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m6, md, true, # links(m6, 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_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; afs_rmdir#(; p_inode0, p_inode1, dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = cache_dirs2 [m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, # links(m6, cache_dirs2), m1, entries0 -- str)).meta:= md ] [entries0[str], (entries0[str] = m6 ∧ 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) ⊕ icache ∧ (afs_asynced ↔ asynced0) ∧ (m6 ∈ cache_dirs2 ⊃ mkinode (m6, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs2), m1, entries0 -- str)).meta, true, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs2), m1, entries0 -- str)).nlink, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs2), m1, entries0 -- str)).nsubdirs, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs2), m1, entries0 -- str)).size) ;mkinode(m6, cache_files[m6].meta, false, cache_files[m6].nlink, 0, cache_files[m6].size) ) = p_inode0 ∧ (entries0[str] ∈ cache_dirs3 ⊃ mkinode (entries0[str], md0, true, (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).nlink, (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).nsubdirs, (entries0[str] = m6 ⊃ 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: ⊦ mkinode(m, md, boolvar, m0, m1, m2).meta = md; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (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)% ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)