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Proof step 122 from proof refine-rmdir (previous step)

# 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, ∅), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, ino = mkdentry(str, entries0[str]).ino, p_ino = m6, mkdentry(str, entries0[str]).dentry?, 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_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size:= (cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size - 1) ] [p_ino ] .nsubdirs ≥ 1, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size ≥ 1, 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 ⊦ ⟨dirs := cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size:= (cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size - 1) ] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size:= ( cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size - 1) ] [p_ino ] .nsubdirs:= ( cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size:= ( cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino ] .size - 1) ] [p_ino ] .nsubdirs - 1) ]; 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), entries0[str] = ROOT_INO

Heuristic symbolic execution applied rule assign right

And obtained the following new goals

Proof step 123:
# links(m6, cache_dirs) = cache_dirs[m6].nlink, # links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, m0 + 1 = cache_dirs[m6].size, m1 + 1 = cache_dirs[m6].nsubdirs, cache_dirs[entries0[str]].entries = ∅, (entries0[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)) = mkdir(m, md4, m8, m9, entries1), (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), (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), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md3, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), cache_dirs[m6] = mkdir(m0 + 1, md1, cache_dirs[m6].nlink, m1 + 1, entries0), entries0[str] = cache_dirs[m6].entries[str], afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size:= (cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size - 1) ] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size:= ( cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size - 1) ] [p_ino ] .nsubdirs:= ( cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size:= ( cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size - 1) ] [p_ino ] .nsubdirs - 1) ], ino = mkdentry(str, entries0[str]).ino, p_ino = m6, mkdentry(str, entries0[str]).dentry?, 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), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), 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), dcache-cons(cache_dcache, 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), icache-cons(cache_icache, 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), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), 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), m6 ∈ cache_dirs, entries0[str] ∈ cache_dirs, str ∈ cache_dirs[m6].entries, str ∈ entries0, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size:= (cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size - 1) ] [p_ino ] .nsubdirs ≥ 1, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size ≥ 1, 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), cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ ⟨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), entries0[str] = ROOT_INO

Rule argument

Additional information

Used simplifier rules: ⊦ mkdentry(str, m).name = str; (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)% ⊦ (ad.entries:= entries).size = ad.size; (s)% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)% ⊦ (ad.size:= m).nsubdirs = ad.nsubdirs; (s)% ⊦ (ad.entries:= entries).nsubdirs = ad.nsubdirs; (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ mkdentry(str, m).dentry?; (s)