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Proof step 68 from proof inv-rename-dir-dentry (previous step)

# links(entries1[str], dirs0) = dirs0[entries1[str]].nlink, # links(m0, dirs0) = dirs0[m0].nlink, # links(m, dirs0) = dirs0[m].nlink, # links(entries[str0], dirs0) = dirs0[entries[str0]].nlink, dirs0[entries1[str]].entries = ∅, dirs0[m0].size = m1 + 1, dirs0[m0] = mkdir(dirs0[m0].size, md4, dirs0[m0].nlink, dirs0[m0].nsubdirs, entries), dirs0[m] = mkdir(dirs0[m].size, md7, dirs0[m].nlink, dirs0[m].nsubdirs, entries1), dirs0[entries1[str]] = mkdir(dirs0[entries1[str]].size, md6, dirs0[entries1[str]].nlink, dirs0[entries1[str]].nsubdirs, ∅), dirs0[entries[str0]] = mkdir(dirs0[entries[str0]].size, md3, dirs0[entries[str0]].nlink, dirs0[entries[str0]].nsubdirs, entries2), entries1[str] = dirs0[m].entries[str], entries[str0] = dirs0[m0].entries[str0], dirs = dirs0[m0, ((dirs0[m0].entries:= (dirs0[m0].entries -- str0)).size:= m1).nsubdirs:= (dirs0[m0].nsubdirs - 1)] [m, (m = m0 ⊃ ((dirs0[m0].entries:= (dirs0[m0].entries -- str0)).size:= m1).nsubdirs:= (dirs0[m0].nsubdirs - 1);dirs0[m]) .entries:= (m = m0 ⊃ ((dirs0[m0].entries:= (dirs0[m0].entries -- str0)).size:= m1).nsubdirs:= (dirs0[m0].nsubdirs - 1);dirs0[m]).entries [str, entries[str0] ] ], afs-cons(dirs0, files), dcache-cons(dcache, dirs0, files), icache-cons(icache, dirs0, files), pcache-cons(pcache, dirs0, files), synced(dirs0, icache, pcache), str ∈ entries1, m0 ∈ dirs0, m ∈ dirs0, entries[str0] ∈ dirs0, entries1[str] ∈ dirs0, str0 ∈ entries, str ∈ dirs0[m].entries, str0 ∈ dirs0[m0].entries, 1 ≤ dirs0[m0].nsubdirs, 1 ≤ dirs0[m0].size, m = m0 → md1 = md2, asynced2 → synced(files, icache, pcache), links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries1[str]).ino; dirs, files); old_dent := mkdentry(str0, entries[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries1[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m0, md2, true, dirs0[m0].nlink, dirs0[m0].nsubdirs, dirs0[m0].size); dirs, files); afs_inode_sync#(mkinode(m, md1, true, # links(m, dirs0), dirs0[m].nsubdirs, dirs0[m].size); dirs, files); new_child_inode := mkinode(entries1[str], md, true, # links(entries1[str], dirs0), dirs0[entries1[str]].nsubdirs, dirs0[entries1[str]].size); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, 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, files[new_child_inode.ino].meta, false, files[new_child_inode.ino].nlink, 0, files[new_child_inode.ino].size) )}; old_parent_inode := (m0 ∈ dirs ⊃ mkinode(m0, dirs[m0].meta, true, dirs[m0].nlink, dirs[m0].nsubdirs, dirs[m0].size) ;mkinode(m0, files[m0].meta, false, files[m0].nlink, 0, files[m0].size) ); new_parent_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, files[m].meta, false, files[m].nlink, 0, files[m].size) ); err := ESUCCESS; asynced := ?] ( afs-cons(dirs, files) → ⟪if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, icache); icache_set#(new_parent_inode, asynced; err, icache); if true then icache_set#(new_child_inode, asynced; err, icache); dcache_delete#(old_parent_inode.ino, old_dent.name; err, dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, dcache)}⟫ ( icache-cons(icache, dirs, files) ∧ dcache-cons(dcache, dirs, files) ∧ pcache-cons(pcache, dirs, files) ∧ synced(dirs, icache, pcache) ∧ (asynced2 → synced(files, icache, pcache)))), dirs0[m0].nsubdirs = 0, dirs0[m0].size = 0, entries1[str] = ROOT_INO, m0 = m ∧ mkdentry(str, entries1[str]) = mkdentry(str0, entries[str0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 69:
# links(entries1[str], dirs0) = dirs0[entries1[str]].nlink, # links(m0, dirs0) = dirs0[m0].nlink, # links(m, dirs0) = dirs0[m].nlink, # links(entries[str0], dirs0) = dirs0[entries[str0]].nlink, dirs0[entries1[str]].entries = ∅, dirs0[m0].size = m1 + 1, dirs0[m0] = mkdir(m1 + 1, md4, dirs0[m0].nlink, dirs0[m0].nsubdirs, entries), dirs0[m] = mkdir(dirs0[m].size, md7, dirs0[m].nlink, dirs0[m].nsubdirs, entries1), dirs0[entries1[str]] = mkdir(dirs0[entries1[str]].size, md6, dirs0[entries1[str]].nlink, dirs0[entries1[str]].nsubdirs, ∅), dirs0[entries[str0]] = mkdir(dirs0[entries[str0]].size, md3, dirs0[entries[str0]].nlink, dirs0[entries[str0]].nsubdirs, entries2), entries1[str] = dirs0[m].entries[str], entries[str0] = dirs0[m0].entries[str0], dirs = dirs0[m0, ((dirs0[m0].entries:= (dirs0[m0].entries -- str0)).size:= m1).nsubdirs:= (dirs0[m0].nsubdirs - 1)] [m, (m = m0 ⊃ ((dirs0[m0].entries:= (dirs0[m0].entries -- str0)).size:= m1).nsubdirs:= (dirs0[m0].nsubdirs - 1);dirs0[m]) .entries:= (m = m0 ⊃ ((dirs0[m0].entries:= (dirs0[m0].entries -- str0)).size:= m1).nsubdirs:= (dirs0[m0].nsubdirs - 1);dirs0[m]).entries [str, entries[str0] ] ], dirs0[m0].nsubdirs ≠ 0, entries1[str] ≠ ROOT_INO, afs-cons(dirs0, files), dcache-cons(dcache, dirs0, files), icache-cons(icache, dirs0, files), pcache-cons(pcache, dirs0, files), synced(dirs0, icache, pcache), str ∈ entries1, m0 ∈ dirs0, m ∈ dirs0, entries[str0] ∈ dirs0, entries1[str] ∈ dirs0, str0 ∈ entries, str ∈ dirs0[m].entries, str0 ∈ dirs0[m0].entries, 1 ≤ dirs0[m0].nsubdirs, m = m0 → md1 = md2, asynced2 → synced(files, icache, pcache), links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries1[str]).ino; dirs, files); old_dent := mkdentry(str0, entries[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries1[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m0, md2, true, dirs0[m0].nlink, dirs0[m0].nsubdirs, m1 + 1); dirs, files); afs_inode_sync#(mkinode(m, md1, true, # links(m, dirs0), dirs0[m].nsubdirs, dirs0[m].size); dirs, files); new_child_inode := mkinode(entries1[str], md, true, # links(entries1[str], dirs0), dirs0[entries1[str]].nsubdirs, dirs0[entries1[str]].size); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, 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, files[new_child_inode.ino].meta, false, files[new_child_inode.ino].nlink, 0, files[new_child_inode.ino].size) )}; old_parent_inode := (m0 ∈ dirs ⊃ mkinode(m0, dirs[m0].meta, true, dirs[m0].nlink, dirs[m0].nsubdirs, dirs[m0].size) ;mkinode(m0, files[m0].meta, false, files[m0].nlink, 0, files[m0].size) ); new_parent_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, files[m].meta, false, files[m].nlink, 0, files[m].size) ); err := ESUCCESS; asynced := ?] ( afs-cons(dirs, files) → ⟪if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, icache); icache_set#(new_parent_inode, asynced; err, icache); if true then icache_set#(new_child_inode, asynced; err, icache); dcache_delete#(old_parent_inode.ino, old_dent.name; err, dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, dcache)}⟫ ( icache-cons(icache, dirs, files) ∧ dcache-cons(dcache, dirs, files) ∧ pcache-cons(pcache, dirs, files) ∧ synced(dirs, icache, pcache) ∧ (asynced2 → synced(files, icache, pcache)))), m0 = m ∧ mkdentry(str, entries1[str]) = mkdentry(str0, entries[str0])

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ m + n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat