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Proof step 163 from proof refine-rename-dir-negdentry (previous step)

# links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, # links(m16, cache_dirs) = cache_dirs[m16].nlink, # links(m9, cache_dirs) = cache_dirs[m9].nlink, m1 + 1 = cache_dirs[m9].nsubdirs, m0 + 1 = cache_dirs[m9].size, (m16 = m9 ⊃ mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str);cache_dirs[m16]) = mkdir(m2, md2, m4, m5, entries), (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, cache_dirs[m9].nlink, m1, entries0 -- str)) = mkdir(m, md, m11, m12, entries2), (m16 = m9 ⊃ mkdir(m2 + 1, md0, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]])) = mkdir(m13, md7, m14, m15, entries4), asynced0 ↔ false, asynced1 ↔ false, entries0[str] = cache_dirs[m9].entries[str], cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md8, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, entries5), cache_dirs[m16] = mkdir(cache_dirs[m16].size, md6, cache_dirs[m16].nlink, cache_dirs[m16].nsubdirs, entries3), cache_dirs[m9] = mkdir(m0 + 1, md3, cache_dirs[m9].nlink, m1 + 1, entries0), dirs = dirs0[old_ino, dirs0[old_ino].size:= (dirs0[old_ino].size - 1)], dirs0 = cache_dirs[old_ino, cache_dirs[old_ino].entries:= (cache_dirs[old_ino].entries -- mkdentry(str, entries0[str]).name)], new_ino = m16, old_dent = mkdentry(str, entries0[str]), old_ino = m9, old_dent.dentry?, afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons (cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], cache_files), dcache-cons ((cache_dcache -- dentrykey(m9, str))[dentrykey(m16, str0), mkdentry(str0, entries0[str])], cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons (cache_icache[m9, mkientry(false, mkinode(m9, md, true, m11, m12, m))][m16, mkientry(false, mkinode(m16, md5, true, m14, m15, m13))], cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced (cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], cache_icache[m9, mkientry(false, mkinode(m9, md, true, m11, m12, m))][m16, mkientry(false, mkinode(m16, md5, true, m14, m15, m13))], cache_pcache), synced(cache_dirs, cache_icache, cache_pcache), str ∈ cache_dirs[m9].entries, entries0[str] ∈ cache_dirs, m9 ∈ cache_dirs, m16 ∈ cache_dirs, str ∈ entries0, old_dent.ino ∈ dirs, dirs[old_ino].nsubdirs ≥ 1, dirs0[old_ino].size ≥ 1, cache_sync → synced (cache_files, cache_icache[m9, mkientry(false, mkinode(m9, md, true, m11, m12, m))][m16, mkientry(false, mkinode(m16, md5, true, m14, m15, m13))], cache_pcache), cache_sync → synced(cache_files, cache_icache, cache_pcache), m16 = m9 → md5 = md0, links(m9, cache_dirs) ≠ ∅ ∨ m9 = ROOT_INO, links(m16, cache_dirs) ≠ ∅ ∨ m16 = ROOT_INO ⊦ ⟨dirs := dirs[old_ino, dirs[old_ino].nsubdirs:= (dirs[old_ino].nsubdirs - 1)] [new_ino, dirs[old_ino, dirs[old_ino].nsubdirs:= (dirs[old_ino].nsubdirs - 1)][new_ino] .entries:= dirs[old_ino, dirs[old_ino].nsubdirs:= (dirs[old_ino].nsubdirs - 1)][new_ino].entries[str0, old_dent.ino] ]; new_dent := negdentry(str0); afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; if asynced1 ∧ ¬ asynced0 then let ino = new_dent.ino in afs_drop_nlink#(ino; dirs, afs_files); if ¬ asynced1 then {dirs[new_ino].size := dirs[new_ino].size + 1; if old_dent.ino ∈ dirs then dirs[new_ino].nsubdirs := dirs[new_ino].nsubdirs + 1}; new_dent := mkdentry(new_dent.name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m9, md0, true, # links(m9, cache_dirs), m1 + 1, m0 + 1); dirs, afs_files); afs_inode_sync#(mkinode(m16, md5, true, # links(m16, cache_dirs), cache_dirs[m16].nsubdirs, cache_dirs[m16].size); dirs, afs_files); new_child_inode := new_child_inode0; if asynced1 then {afs_inode_sync#(new_child_inode; dirs, afs_files); new_child_inode := (new_child_inode.ino ∈ dirs ⊃ mkinode (new_child_inode.ino, dirs[new_child_inode.ino].meta, true, dirs[new_child_inode.ino].nlink, dirs[new_child_inode.ino].nsubdirs, dirs[new_child_inode.ino].size) ;mkinode (new_child_inode.ino, afs_files[new_child_inode.ino].meta, false, afs_files[new_child_inode.ino].nlink, 0, afs_files[new_child_inode.ino].size) )}; old_parent_inode := (m9 ∈ dirs ⊃ mkinode(m9, dirs[m9].meta, true, dirs[m9].nlink, dirs[m9].nsubdirs, dirs[m9].size) ;mkinode(m9, afs_files[m9].meta, false, afs_files[m9].nlink, 0, afs_files[m9].size) ); new_parent_inode := (m16 ∈ dirs ⊃ mkinode(m16, dirs[m16].meta, true, dirs[m16].nlink, dirs[m16].nsubdirs, dirs[m16].size) ;mkinode(m16, afs_files[m16].meta, false, afs_files[m16].nlink, 0, afs_files[m16].size) ); err := ESUCCESS; asynced := ?⟩ ( mkinode (m9, (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, # links(m9, cache_dirs), m1, entries0 -- str)) .meta, true, (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, # links(m9, cache_dirs), m1, entries0 -- str)) .nlink, (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, # links(m9, cache_dirs), m1, entries0 -- str)) .nsubdirs, (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, # links(m9, cache_dirs), m1, entries0 -- str)) .size) = old_parent_inode ∧ mkinode(m16, md5, true, m14, m15, m13) = new_parent_inode ∧ new_child_inode0 = new_child_inode ∧ negdentry(str) = old_dent ∧ mkdentry(str0, entries0[str]) = new_dent ∧ ESUCCESS = err ∧ dirs = cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]) ;mkdir(m0, md3, # links(m9, cache_dirs), m1, entries0 -- str) ) .meta:= md0 ] [m16, (m16 = m9 ⊃ mkdir(m2 + 1, md0, m4, m5 + 1, entries[str0, entries0[str]]) ;mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]) ) .meta:= md5 ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[m9, mkientry(false, mkinode(m9, md, true, m11, m12, m))][m16, mkientry(false, mkinode(m16, md5, true, m14, m15, m13))] ∧ (asynced ↔ asynced2)), str0 ∈ entries3, str0 ∈ cache_dirs[m16].entries

Heuristic symbolic execution applied rule assign right

And obtained the following new goals

Proof step 164:
# links(m9, cache_dirs) = cache_dirs[m9].nlink, # links(m16, cache_dirs) = cache_dirs[m16].nlink, # links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, m0 + 1 = cache_dirs[m9].size, m1 + 1 = cache_dirs[m9].nsubdirs, (m16 = m9 ⊃ mkdir(m2 + 1, md0, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]])) = mkdir(m13, md7, m14, m15, entries4), (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, cache_dirs[m9].nlink, m1, entries0 -- str)) = mkdir(m, md, m11, m12, entries2), (m16 = m9 ⊃ mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str);cache_dirs[m16]) = mkdir(m2, md2, m4, m5, entries), asynced1 ↔ false, asynced0 ↔ false, cache_dirs[m9] = mkdir(m0 + 1, md3, cache_dirs[m9].nlink, m1 + 1, entries0), cache_dirs[m16] = mkdir(cache_dirs[m16].size, md6, cache_dirs[m16].nlink, cache_dirs[m16].nsubdirs, entries3), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md8, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, entries5), entries0[str] = cache_dirs[m9].entries[str], dirs = dirs1[old_ino, dirs1[old_ino].nsubdirs:= (dirs1[old_ino].nsubdirs - 1)] [new_ino, dirs1[old_ino, dirs1[old_ino].nsubdirs:= (dirs1[old_ino].nsubdirs - 1)][new_ino] .entries:= dirs1[old_ino, dirs1[old_ino].nsubdirs:= (dirs1[old_ino].nsubdirs - 1)][new_ino].entries[str0, old_dent.ino] ], dirs0 = cache_dirs[old_ino, cache_dirs[old_ino].entries:= (cache_dirs[old_ino].entries -- mkdentry(str, entries0[str]).name)], dirs1 = dirs0[old_ino, dirs0[old_ino].size:= (dirs0[old_ino].size - 1)], new_ino = m16, old_dent = mkdentry(str, entries0[str]), old_ino = m9, old_dent.dentry?, afs-cons (cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], 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), dcache-cons ((cache_dcache -- dentrykey(m9, str))[dentrykey(m16, str0), mkdentry(str0, entries0[str])], cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], cache_files), icache-cons(cache_icache, cache_dirs, cache_files), icache-cons (cache_icache[m9, mkientry(false, mkinode(m9, md, true, m11, m12, m))][m16, mkientry(false, mkinode(m16, md5, true, m14, m15, m13))], cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], cache_files), synced(cache_dirs, cache_icache, cache_pcache), synced (cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md3, cache_dirs[m9].nlink, m1, entries0 -- str)).meta:= md0 ] [m16, mkdir(m13, md5, m14, m15, entries4) ], cache_icache[m9, mkientry(false, mkinode(m9, md, true, m11, m12, m))][m16, mkientry(false, mkinode(m16, md5, true, m14, m15, m13))], cache_pcache), old_dent.ino ∈ dirs1, str ∈ entries0, m16 ∈ cache_dirs, m9 ∈ cache_dirs, entries0[str] ∈ cache_dirs, str ∈ cache_dirs[m9].entries, dirs0[old_ino].size ≥ 1, dirs1[old_ino].nsubdirs ≥ 1, m16 = m9 → md5 = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), cache_sync → synced (cache_files, cache_icache[m9, mkientry(false, mkinode(m9, md, true, m11, m12, m))][m16, mkientry(false, mkinode(m16, md5, true, m14, m15, m13))], cache_pcache), links(m16, cache_dirs) ≠ ∅ ∨ m16 = ROOT_INO, links(m9, cache_dirs) ≠ ∅ ∨ m9 = ROOT_INO ⊦ ⟨new_dent := negdentry(str0); afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; if asynced1 ∧ ¬ asynced0 then let ino = new_dent.ino in afs_drop_nlink#(ino; dirs, afs_files); if ¬ asynced1 then {dirs[new_ino].size := dirs[new_ino].size + 1; if old_dent.ino ∈ dirs then dirs[new_ino].nsubdirs := dirs[new_ino].nsubdirs + 1}; new_dent := mkdentry(new_dent.name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m9, md0, true, # links(m9, cache_dirs), m1 + 1, m0 + 1); dirs, afs_files); afs_inode_sync#(mkinode(m16, md5, true, # links(m16, cache_dirs), cache_dirs[m16].nsubdirs, cache_dirs[m16].size); dirs, afs_files); new_child_inode := new_child_inode0; if asynced1 then {afs_inode_sync#(new_child_inode; dirs, afs_files); new_child_inode := (new_child_inode.ino ∈ dirs ⊃ mkinode (new_child_inode.ino, dirs[new_child_inode.ino].meta, true, dirs[new_child_inode.ino].nlink, dirs[new_child_inode.ino].nsubdirs, dirs[new_child_inode.ino].size) ;mkinode (new_child_inode.ino, afs_files[new_child_inode.ino].meta, false, afs_files[new_child_inode.ino].nlink, 0, afs_files[new_child_inode.ino].size) )}; old_parent_inode := (m9 ∈ dirs ⊃ mkinode(m9, dirs[m9].meta, true, dirs[m9].nlink, dirs[m9].nsubdirs, dirs[m9].size) ;mkinode(m9, afs_files[m9].meta, false, afs_files[m9].nlink, 0, afs_files[m9].size) ); new_parent_inode := (m16 ∈ dirs ⊃ mkinode(m16, dirs[m16].meta, true, dirs[m16].nlink, dirs[m16].nsubdirs, dirs[m16].size) ;mkinode(m16, afs_files[m16].meta, false, afs_files[m16].nlink, 0, afs_files[m16].size) ); err := ESUCCESS; asynced := ?⟩ ( mkinode (m9, (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, # links(m9, cache_dirs), m1, entries0 -- str)) .meta, true, (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, # links(m9, cache_dirs), m1, entries0 -- str)) .nlink, (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, # links(m9, cache_dirs), m1, entries0 -- str)) .nsubdirs, (m9 = m16 ⊃ mkdir(m2 + 1, md5, m4, m5 + 1, entries[str0, entries0[str]]);mkdir(m0, md0, # links(m9, cache_dirs), m1, entries0 -- str)) .size) = old_parent_inode ∧ mkinode(m16, md5, true, m14, m15, m13) = new_parent_inode ∧ new_child_inode0 = new_child_inode ∧ negdentry(str) = old_dent ∧ mkdentry(str0, entries0[str]) = new_dent ∧ ESUCCESS = err ∧ dirs = cache_dirs [m9, (m9 = m16 ⊃ mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]) ;mkdir(m0, md3, # links(m9, cache_dirs), m1, entries0 -- str) ) .meta:= md0 ] [m16, (m16 = m9 ⊃ mkdir(m2 + 1, md0, m4, m5 + 1, entries[str0, entries0[str]]) ;mkdir(m2 + 1, md2, m4, m5 + 1, entries[str0, entries0[str]]) ) .meta:= md5 ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[m9, mkientry(false, mkinode(m9, md, true, m11, m12, m))][m16, mkientry(false, mkinode(m16, md5, true, m14, m15, m13))] ∧ (asynced ↔ asynced2)), str0 ∈ cache_dirs[m16].entries, str0 ∈ entries3

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).ino = m; (s)% ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (s)% ⊦ mkdentry(str, m).dentry?; (s)