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Proof step 21 from proof inv-rmdir (previous step)

dirs0[m] = mkdir(m1, md2, # links(m, dirs0), m4, entries), dirs0[entries[str]] = mkdir(m0, md1, # links(entries[str], dirs0), m2, ∅), c_inode = mkinode(entries[str], md, true, # links(entries[str], dirs0), m2, m0), dent = mkdentry(str, entries[str]), dirs = dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)] [p_ino, dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino] .size:= (dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino].size - 1) ] [p_ino, dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)] [p_ino, dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino] .size:= (dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino].size - 1) ] [p_ino ] .nsubdirs:= ( dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)] [p_ino, dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino] .size:= (dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino].size - 1) ] [p_ino ] .nsubdirs - 1) ], ino = dent.ino, p_ino = p_inode.ino, p_inode = mkinode(m, md0, true, # links(m, dirs0), m4, m1), dent.dentry?, afs-cons(dirs0, files), pre-rmdir(p_inode, c_inode, dent, dirs0, files), m ∈ dirs0, str ∈ entries, entries[str] ∈ dirs0, dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)] [p_ino, dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino] .size:= (dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino].size - 1) ] [p_ino ] .nsubdirs ≥ 1, dirs0[p_ino, dirs0[p_ino].entries:= (dirs0[p_ino].entries -- dent.name)][p_ino].size ≥ 1, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪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 := ?⟫ afs-cons(dirs, files), entries[str] = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 22:
# links(entries[str], dirs0) = dirs0[entries[str]].nlink, # links(m, dirs0) = dirs0[m].nlink, dirs0[entries[str]].entries = ∅, entries[str] = dirs0[m].entries[str], dirs0[m] = mkdir(dirs0[m].size, md2, dirs0[m].nlink, dirs0[m].nsubdirs, entries), dirs0[entries[str]] = mkdir(dirs0[entries[str]].size, md1, dirs0[entries[str]].nlink, dirs0[entries[str]].nsubdirs, ∅), dirs = dirs0[m, ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1)).nsubdirs:= (dirs0[m].nsubdirs - 1)], dirs0[m].nsubdirs ≠ 0, dirs0[m].size ≠ 0, entries[str] ≠ ROOT_INO, afs-cons(dirs0, files), str ∈ dirs0[m].entries, m ∈ dirs0, str ∈ entries, entries[str] ∈ dirs0, 1 ≤ dirs0[m].nsubdirs, 1 ≤ dirs0[m].size, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(m, md0, true, dirs0[m].nlink, dirs0[m].nsubdirs, dirs0[m].size); dirs, files); afs_inode_sync# (mkinode(entries[str], md, true, # links(entries[str], dirs0), dirs0[entries[str]].nsubdirs, dirs0[entries[str]].size); dirs, files); p_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) ); c_inode := getinode(entries[str], dirs, files); err := ESUCCESS; asynced := ?⟫ afs-cons(dirs, files)

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%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat%⊦ 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)% ⊦ 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)% ⊦ 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)% ⊦ 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)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ (ad.entries:= entries).nsubdirs = ad.nsubdirs; (s)% ⊦ (ad.size:= m).nsubdirs = ad.nsubdirs; (s)% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)% ⊦ (ad.entries:= entries).size = ad.size; (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)% ⊦ mkdentry(str, m).name = str; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)