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Proof step 115 from proof refine-rmdir (previous step)

(m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries0 -- str)) = mkdir(m10, md5, m11, m12, entries2), (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), (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)) = mkdir(m, md4, m8, m9, entries1), 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, c_inode = p_inode1, cache_dirs2 = cache_dirs, dent = dent0, dent0 = mkdentry(str, entries0[str]), dirs = cache_dirs, err = err0, files = afs_files, p_inode = p_inode0, 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), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), 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], mkdir(m, md0, m8, m9, entries1) ], cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), dcache-cons (cache_dcache -- dentrykey(m6, 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], mkdir(m, md0, m8, m9, entries1) ], cache_files), icache-cons(cache_icache, cache_dirs, cache_files), icache-cons (cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], 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], mkdir(m, md0, m8, m9, entries1) ], cache_files), pcache-cons(cache_pcache, cache_dirs, 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], mkdir(m, md0, m8, m9, entries1) ], cache_files), pre-rmdir(p_inode, c_inode, dent, dirs, 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], mkdir(m, md0, m8, m9, entries1) ], cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_pcache), 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), cache_sync → synced (cache_files, cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ ( 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 ] ∧ files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ] ∧ (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_inode ∧ (entries0[str] ∈ cache_dirs2 ⊃ mkinode(entries0[str], md0, true, m8, m9, m) ;mkinode(entries0[str], cache_files[entries0[str]].meta, false, cache_files[entries0[str]].nlink, 0, cache_files[entries0[str]].size) ) = c_inode ∧ negdentry(str) = dent ∧ ESUCCESS = err), ⟨let p_ino = p_inode.ino in {dirs[p_ino].entries := dirs[p_ino].entries -- dent.name; dirs[p_ino].size := dirs[p_ino].size - 1; dirs[p_ino].nsubdirs := dirs[p_ino].nsubdirs - 1; let ino = dent.ino in afs_drop_nlink#(ino; dirs, files); dent := negdentry(dent.name); afs_inode_sync#(p_inode; dirs, files); afs_inode_sync#(c_inode; dirs, files); p_inode := getinode(p_inode.ino, dirs, files); c_inode := getinode(c_inode.ino, dirs, files); err := ESUCCESS}; asynced := ?⟩ ( 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 ] ∧ files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ] ∧ (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_inode ∧ (entries0[str] ∈ cache_dirs2 ⊃ mkinode(entries0[str], md0, true, m8, m9, m) ;mkinode(entries0[str], cache_files[entries0[str]].meta, false, cache_files[entries0[str]].nlink, 0, cache_files[entries0[str]].size) ) = c_inode ∧ negdentry(str) = dent ∧ ESUCCESS = err), entries0[str] = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 116:
# links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, # links(m6, cache_dirs) = cache_dirs[m6].nlink, m1 + 1 = cache_dirs[m6].nsubdirs, m0 + 1 = cache_dirs[m6].size, cache_dirs[entries0[str]].entries = ∅, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, cache_dirs[m6].nlink, m1, entries0 -- str)) = mkdir(m10, md5, m11, m12, entries2), (entries0[str] = m6 ⊃ mkdir(m0, md1, cache_dirs[entries0[str]].nlink, m1, entries0 -- str);cache_dirs[entries0[str]]) = mkdir(m5, md2, m3 + 1, m7, entries), (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)) = mkdir(m, md4, m8, m9, entries1), entries0[str] = cache_dirs[m6].entries[str], cache_dirs[m6] = mkdir(m0 + 1, md1, cache_dirs[m6].nlink, m1 + 1, entries0), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md3, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), entries0[str] ≠ ROOT_INO, afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), afs-cons (cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, cache_dirs[m6].nlink, m1, entries0 -- str)).meta:= md] [entries0[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), dcache-cons (cache_dcache -- dentrykey(m6, str), cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, cache_dirs[m6].nlink, m1, entries0 -- str)).meta:= md] [entries0[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), icache-cons(cache_icache, cache_dirs, cache_files), icache-cons (cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, cache_dirs[m6].nlink, m1, entries0 -- str)).meta:= md] [entries0[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, cache_dirs[m6].nlink, m1, entries0 -- str)).meta:= md] [entries0[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), synced (cache_dirs[m6, (m6 = entries0[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md1, cache_dirs[m6].nlink, m1, entries0 -- str)).meta:= md] [entries0[str], mkdir(m, md0, m8, m9, entries1) ], cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_pcache), synced(cache_dirs, cache_icache, cache_pcache), str ∈ cache_dirs[m6].entries, entries0[str] ∈ cache_dirs, m6 ∈ cache_dirs, str ∈ entries0, cache_sync → synced(cache_files, cache_icache, cache_pcache), cache_sync → synced (cache_files, cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ ⟨let p_ino = m6 in {dirs := cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)]; dirs[p_ino].size := dirs[p_ino].size - 1; dirs[p_ino].nsubdirs := dirs[p_ino].nsubdirs - 1; let ino = mkdentry(str, entries0[str]).ino in {afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; afs_drop_nlink#(ino; dirs, afs_files)}; dent := negdentry(mkdentry(str, entries0[str]).name); afs_inode_sync#(mkinode(m6, md, true, # links(m6, cache_dirs), m1 + 1, m0 + 1); dirs, afs_files); afs_inode_sync# (mkinode(entries0[str], md0, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size) ; dirs, afs_files); p_inode := (m6 ∈ dirs ⊃ mkinode(m6, dirs[m6].meta, true, dirs[m6].nlink, dirs[m6].nsubdirs, dirs[m6].size) ;mkinode(m6, afs_files[m6].meta, false, afs_files[m6].nlink, 0, afs_files[m6].size) ); c_inode := getinode(entries0[str], dirs, afs_files); err := ESUCCESS}; asynced := ?⟩ ( dirs = 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 ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[entries0[str], mkientry(false, mkinode(entries0[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ] ∧ (asynced ↔ asynced0) ∧ mkinode (m6, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries0 -- str)).meta, true, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries0 -- str)).nlink, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries0 -- str)).nsubdirs, (m6 = entries0[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries0 -- str)).size) = p_inode ∧ mkinode(entries0[str], md0, true, m8, m9, m) = c_inode ∧ negdentry(str) = dent ∧ ESUCCESS = err)

Rule argument

Additional information

Used simplifier rules: afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base%⊦ 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)% ⊦ mkdentry(str, m) ≠ negdentry(str0); (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).directory ↔ boolvar; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nsubdirs = m1; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).size = m2; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nlink = m0; (s)% ⊦ valid-dir-ino(ino, dirs, files) ↔ ino ∈ dirs; (s)%⊦ valid-dir-inode(inode, dirs, files) ↔ valid-dir-ino(inode.ino, dirs, files) ∧ inode.nlink = dirs[inode.ino].nlink ∧ inode.size = dirs[inode.ino].size ∧ inode.nsubdirs = dirs[inode.ino].nsubdirs ∧ inode.directory; (s)% ⊦ dir-empty(ino, dirs, files) ↔ dirs[ino].entries = ∅; (s)% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ mkdentry(str, m).name = str; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (s)%⊦ valid-dentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ valid-ino(dent.ino, dirs, files) ∧ dent.name ∈ dirs[ino].entries ∧ dent.ino = dirs[ino].entries[dent.name] ∧ dent.dentry?; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)% ⊦ mkdentry(str, m).ino = m; (s)%⊦ pre-rmdir(p_inode, c_inode, dent, dirs, files) ↔ dent.ino ≠ ROOT_INO ∧ valid-dentry(p_inode.ino, dent, dirs, files) ∧ dir-empty(dent.ino, dirs, files) ∧ valid-dir-inode(p_inode, dirs, files) ∧ valid-dir-inode(c_inode, dirs, files) ∧ c_inode.ino = dent.ino; (s)