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Proof step 112 from proof refine-rename-file-negdentry (previous step)

# links (entries[str], cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)).meta:= md0 ] [m3, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).meta:= md5 ]) = # links(entries[str], cache_dirs), (m3 = m8 ⊃ mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str);mkdir(m7, md7, # links(m3, cache_dirs), m11, entries2)) = mkdir(m0, md6, m4, m6, entries1), cache_files[entries[str]] = mkfile(m2, md3, # links(entries[str], cache_dirs), content0), cache_dirs[m3] = mkdir(m7, md7, # links(m3, cache_dirs), m11, entries2), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m9, md2, # links(entries[str], cache_dirs), content), cache_dirs[m8] = mkdir(m1 + 1, md4, # links(m8, cache_dirs), m5, entries), cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)).meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] [m8 ] = mkdir(m, md, m12, m13, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = ESUCCESS, afs-cons (cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)).meta:= md0 ] [m3, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).meta:= md5 ], 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), m8 ∈ cache_dirs, m3 ∈ cache_dirs, entries[str] ∈ cache_files, str ∈ entries, m3 = m8 → md5 = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m8, cache_dirs) ≠ ∅ ∨ m8 = ROOT_INO ⊦ ⟪icache_set#(mkinode(m8, md, true, m12, m13, m), false; err, cache_icache); icache_set# (( m3 ∈ cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] ⊃ mkinode (m3, cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] [m3 ] .meta, true, cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] [m3 ] .nlink, cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] [m3 ] .nsubdirs, cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] [m3 ] .size) ;mkinode(m3, cache_files[m3].meta, false, cache_files[m3].nlink, 0, cache_files[m3].size) ), false ; err, cache_icache); skip; dcache_delete#(m8, mkdentry(str, entries[str]).name; err, cache_dcache); dcache_set#(m3, mkdentry(str0, mkdentry(str, entries[str]).ino).name, mkdentry(str0, mkdentry(str, entries[str]).ino); err, cache_dcache)⟫ ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m3, md5, true, # links(m3, cache_dirs), m11, m7); old_parent_inode0 := mkinode(m8, md0, true, # links(m8, cache_dirs), m5, m1 + 1); old_child_inode0 := mkinode(entries[str], md1, false, # links(entries[str], cache_dirs), 0, m10); cache_dirs0 := cache_dirs; new_child_inode1 := new_child_inode0; cache_dirs1 := cache_dirs; cache_dirs2 := cache_dirs; cache_dirs3 := 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)⟩ ( (m8 ∈ cache_dirs3 ⊃ mkinode (m8, (m8 = m3 ⊃ mkdir(m0 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md0, # links(m8, cache_dirs3), m5, entries -- str)) .meta, true, (m8 = m3 ⊃ mkdir(m0 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md0, # links(m8, cache_dirs3), m5, entries -- str)) .nlink, (m8 = m3 ⊃ mkdir(m0 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md0, # links(m8, cache_dirs3), m5, entries -- str)) .nsubdirs, (m8 = m3 ⊃ mkdir(m0 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md0, # links(m8, cache_dirs3), m5, entries -- str)) .size) ;mkinode(m8, cache_files[m8].meta, false, cache_files[m8].nlink, 0, cache_files[m8].size) ) = old_parent_inode0 ∧ (m3 ∈ cache_dirs3 ⊃ mkinode (m3, md5, true, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).nlink, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).nsubdirs, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).size) ;mkinode(m3, cache_files[m3].meta, false, cache_files[m3].nlink, 0, cache_files[m3].size) ) = new_parent_inode0 ∧ mkinode(entries[str], md1, false, # links(entries[str], cache_dirs0), 0, m10) = old_child_inode0 ∧ new_child_inode1 = new_child_inode0 ∧ negdentry(str) = old_dent0 ∧ mkdentry(str0, entries[str]) = new_dent0 ∧ err = err0 ∧ cache_dirs = cache_dirs3 [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs3), m5, entries -- str) ) .meta:= md0 ] [m3, (m3 = m8 ∧ m8 ∈ cache_dirs3 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]) ;(m3 ∈ cache_dirs3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);cache_dirs3[m3]) ) .meta:= md5 ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced0)), entries[str] ∈ cache_dirs, str0 ∈ entries2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 113:
# links (entries[str], cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)).meta:= md0 ] [m3, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).meta:= md5 ]) = # links(entries[str], cache_dirs), (m3 = m8 ⊃ mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str);mkdir(m7, md7, # links(m3, cache_dirs), m11, entries2)) = mkdir(m0, md6, m4, m6, entries1), (m8 = m3 ⊃ mkdir(m0 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md0, # links(m8, cache_dirs), m5, entries -- str)) = mkdir(m, md, m12, m13, entries0), cache_dirs[m8] = mkdir(m1 + 1, md4, # links(m8, cache_dirs), m5, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m9, md2, # links(entries[str], cache_dirs), content), cache_dirs[m3] = mkdir(m7, md7, # links(m3, cache_dirs), m11, entries2), cache_files[entries[str]] = mkfile(m2, md3, # links(entries[str], cache_dirs), content0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = ESUCCESS, afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons (cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)).meta:= md0 ] [m3, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).meta:= md5 ], 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 ∈ entries, entries[str] ∈ cache_files, m3 ∈ cache_dirs, m8 ∈ cache_dirs, ¬ str0 ∈ entries2, ¬ entries[str] ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), m3 = m8 → md5 = md0, links(m8, cache_dirs) ≠ ∅ ∨ m8 = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ ⟪icache_set#(mkinode(m8, md, true, m12, m13, m), false; err, cache_icache); icache_set# (( m3 ∈ cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] ⊃ mkinode (m3, cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] [m3 ] .meta, true, cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] [m3 ] .nlink, cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] [m3 ] .nsubdirs, cache_dirs [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str)) .meta:= md0 ] [m3, cache_dirs[m3, mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])] [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs), m5, entries -- str) ) .meta:= md0 ] [m3 ] .meta:= md5 ] [m3 ] .size) ;mkinode(m3, cache_files[m3].meta, false, cache_files[m3].nlink, 0, cache_files[m3].size) ), false ; err, cache_icache); skip; dcache_delete#(m8, mkdentry(str, entries[str]).name; err, cache_dcache); dcache_set#(m3, mkdentry(str0, mkdentry(str, entries[str]).ino).name, mkdentry(str0, mkdentry(str, entries[str]).ino); err, cache_dcache)⟫ ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m3, md5, true, # links(m3, cache_dirs), m11, m7); old_parent_inode0 := mkinode(m8, md0, true, # links(m8, cache_dirs), m5, m1 + 1); old_child_inode0 := mkinode(entries[str], md1, false, # links(entries[str], cache_dirs), 0, m10); cache_dirs0 := cache_dirs; new_child_inode1 := new_child_inode0; cache_dirs1 := cache_dirs; cache_dirs2 := cache_dirs; cache_dirs3 := 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)⟩ ( (m8 ∈ cache_dirs3 ⊃ mkinode (m8, (m8 = m3 ⊃ mkdir(m0 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md0, # links(m8, cache_dirs3), m5, entries -- str)) .meta, true, (m8 = m3 ⊃ mkdir(m0 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md0, # links(m8, cache_dirs3), m5, entries -- str)) .nlink, (m8 = m3 ⊃ mkdir(m0 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md0, # links(m8, cache_dirs3), m5, entries -- str)) .nsubdirs, (m8 = m3 ⊃ mkdir(m0 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m1, md0, # links(m8, cache_dirs3), m5, entries -- str)) .size) ;mkinode(m8, cache_files[m8].meta, false, cache_files[m8].nlink, 0, cache_files[m8].size) ) = old_parent_inode0 ∧ (m3 ∈ cache_dirs3 ⊃ mkinode (m3, md5, true, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).nlink, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).nsubdirs, (m3 = m8 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]])).size) ;mkinode(m3, cache_files[m3].meta, false, cache_files[m3].nlink, 0, cache_files[m3].size) ) = new_parent_inode0 ∧ mkinode(entries[str], md1, false, # links(entries[str], cache_dirs0), 0, m10) = old_child_inode0 ∧ new_child_inode1 = new_child_inode0 ∧ negdentry(str) = old_dent0 ∧ mkdentry(str0, entries[str]) = new_dent0 ∧ err = err0 ∧ cache_dirs = cache_dirs3 [m8, (m8 = m3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]) ;mkdir(m1, md4, # links(m8, cache_dirs3), m5, entries -- str) ) .meta:= md0 ] [m3, (m3 = m8 ∧ m8 ∈ cache_dirs3 ⊃ mkdir(m0 + 1, md0, m4, m6, entries1[str0, entries[str]]) ;(m3 ∈ cache_dirs3 ⊃ mkdir(m0 + 1, md6, m4, m6, entries1[str0, entries[str]]);cache_dirs3[m3]) ) .meta:= md5 ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced0))

Rule argument

Additional information

Used simplifier rules: ⊦ 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)