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Proof step 196 from proof refine-rmdir (previous step)

# links(ino1, dirs0) = dirs0[ino1].nlink, # entries = dirs0[ino1].size, # links(entries[str], dirs0) = dirs0[entries[str]].nlink, dirs0[entries[str]].entries = ∅, 0 = dirs0[entries[str]].size, dirs0[entries[str]] = mkdir(0, md0, dirs0[entries[str]].nlink, dirs0[entries[str]].nsubdirs, ∅), dirs0[ino1] = mkdir(# entries, md, dirs0[ino1].nlink, dirs0[ino1].nsubdirs, entries), root[q + str '] = dnode(md1, ∅), entries[str] = dirs0[ino1].entries[str], fs = abs-files(files), dirs0[ino1].nsubdirs ≠ 0, entries ≠ ∅, root.dir?, root[q].dir?, dom of < ino0, abs(root, ROOT_INO)(dirs0), afs-cons(dirs0, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs0, files), of-cons(of, dirs0, files), pw(user, md), px(user, md), px-parent(user, q, ROOT_INO, dirs0, files), str ∈ entries, ino1 ∈ dirs0, entries[str] ∈ dirs0, str ∈ dirs0[ino1].entries, dirs ≡ dirs0[ino1, ((dirs0[ino1].entries:= (dirs0[ino1].entries -- str)).size:= (# entries - 1)).nsubdirs:= (dirs0[ino1].nsubdirs - 1)], 1 ≤ # entries, 1 ≤ dirs0[ino1].nsubdirs, psynced → asynced0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(ino1, md, true, dirs0[ino1].nlink, dirs0[ino1].nsubdirs, # entries); dirs, files); afs_inode_sync#(mkinode(entries[str], md0, true, # links(entries[str], dirs0), dirs0[entries[str]].nsubdirs, 0); dirs, files); inode := (ino1 ∈ dirs ⊃ mkinode(ino1, dirs[ino1].meta, true, dirs[ino1].nlink, dirs[ino1].nsubdirs, dirs[ino1].size) ;mkinode(ino1, files[ino1].meta, false, files[ino1].nlink, 0, files[ino1].size) ); c_inode := getinode(entries[str], dirs, files); err := ESUCCESS; asynced := ?; vfs_put_inode#(c_inode, of, asynced; err, dirs, files)] ( of-cons(of, dirs, files) ∧ afs-cons(dirs, files) → ⟨posix_rmdir#(q + str ', user, fs, abs-of(of); err1, root; psynced)⟩ (fs = abs-files(files) ∧ abs(root, ROOT_INO)(dirs) ∧ (psynced → asynced) ∧ err = err1))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 197:
# links(entries[str], dirs0) = dirs0[entries[str]].nlink, # entries = dirs0[ino1].size, # links(ino1, dirs0) = dirs0[ino1].nlink, # entries = m + 1, dirs0[entries[str]].entries = ∅, 0 = dirs0[entries[str]].size, entries[str] = dirs0[ino1].entries[str], root[q + str '] = dnode(md1, ∅), dirs0[ino1] = mkdir(# entries, md, dirs0[ino1].nlink, dirs0[ino1].nsubdirs, entries), dirs0[entries[str]] = mkdir(0, md0, dirs0[entries[str]].nlink, dirs0[entries[str]].nsubdirs, ∅), fs = abs-files(files), root[q].dir?, root.dir?, dom of < ino0, abs(root, ROOT_INO)(dirs0), afs-cons(dirs0, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs0, files), of-cons(of, dirs0, files), pw(user, md), px(user, md), px-parent(user, q, ROOT_INO, dirs0, files), str ∈ dirs0[ino1].entries, entries[str] ∈ dirs0, ino1 ∈ dirs0, str ∈ entries, dirs ≡ dirs0[ino1, ((dirs0[ino1].entries:= (dirs0[ino1].entries -- str)).size:= m).nsubdirs:= (dirs0[ino1].nsubdirs - 1)], 1 ≤ dirs0[ino1].nsubdirs, 1 ≤ # entries, psynced → asynced0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(ino1, md, true, dirs0[ino1].nlink, dirs0[ino1].nsubdirs, # entries); dirs, files); afs_inode_sync#(mkinode(entries[str], md0, true, # links(entries[str], dirs0), dirs0[entries[str]].nsubdirs, 0); dirs, files); inode := (ino1 ∈ dirs ⊃ mkinode(ino1, dirs[ino1].meta, true, dirs[ino1].nlink, dirs[ino1].nsubdirs, dirs[ino1].size) ;mkinode(ino1, files[ino1].meta, false, files[ino1].nlink, 0, files[ino1].size) ); c_inode := getinode(entries[str], dirs, files); err := ESUCCESS; asynced := ?; vfs_put_inode#(c_inode, of, asynced; err, dirs, files)] ( of-cons(of, dirs, files) ∧ afs-cons(dirs, files) → ⟨posix_rmdir#(q + str ', user, fs, abs-of(of); err1, root; psynced)⟩ (fs = abs-files(files) ∧ abs(root, ROOT_INO)(dirs) ∧ (psynced → asynced) ∧ err = err1)), # entries = 0, dirs0[ino1].nsubdirs = 0, entries = ∅

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [# entries, m]

Additional information

Used simplifier rules: ⊦ # entries = 0 ↔ entries = ∅; (s)%n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: # entries, m