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Proof step 569 from proof refine-rmdir (previous step)

# links(ino1, dirs) = m0, # entries = m, # links(entries[str], dirs) = dirs[entries[str]].nlink, dirs[entries[str]].entries = ∅, 0 = dirs[entries[str]].size, root[q + str '] = dnode(md1, ∅), dirs[ino1] = mkdir(# entries, md, m0, m1, entries), dirs[entries[str]] = mkdir(0, md0, dirs[entries[str]].nlink, dirs[entries[str]].nsubdirs, ∅), entries[str] = entries0[str], dirs[ino1] = mkdir(m, md2, m0, m1, entries0), root[q].dir?, root.dir?, dom of < ino0, abs(root, ROOT_INO)(dirs), afs-cons(dirs, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs, files), of-cons(of, dirs, files), pw(user, md), px(user, md), px-parent(user, q, ROOT_INO, dirs, files), entries[str] ∈ dirs, ino1 ∈ dirs, str ∈ entries, str ∈ entries0, psynced → asynced0, links(ino1, dirs) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ [fail#(; ; err); asynced := ?; files0 := files; vfs_put_inode#(mkinode(entries[str], md0, true, dirs[entries[str]].nlink, dirs[entries[str]].nsubdirs, 0), of, asynced; err, dirs, files)] ( of-cons(of, dirs, files) ∧ afs-cons(dirs, files) → ⟨posix_rmdir#(q + str ', user, abs-files(files0), abs-of(of); err1, root; psynced)⟩ (abs-files(files0) = abs-files(files) ∧ abs(root, ROOT_INO)(dirs) ∧ (psynced → asynced) ∧ err = err1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 570:
⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n, md, n0, n1, entries) → # entries = n
Proof step 571:
⊦ ispath(p, ino, m, dirs, files) ∧ px-parent(user, p, ino, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) ∧ m ∈ dirs ∧ dirs[m] = mkdir(m0, md, n0, n1, entries) ∧ str ∈ entries ∧ entries[str] ∈ dirs → (dirs[entries[str]].entries = ∅ ↔ tree[p + str '].entries = ∅)
Proof step 572:
⊦ ispath(p, ino, m, dirs, files) ∧ px-parent(user, p, ino, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) ∧ m ∈ dirs ∧ dirs[m] = mkdir(m0, md, n0, n1, entries) ∧ str ∈ entries → (entries[str] ∈ dirs ↔ tree[p + str '].dir?)
Proof step 573:
# links(entries0[str], dirs) = dirs[entries0[str]].nlink, dirs[entries0[str]].entries = ∅, 0 = dirs[entries0[str]].size, dirs[entries0[str]] = mkdir(0, md0, dirs[entries0[str]].nlink, dirs[entries0[str]].nsubdirs, ∅), dirs[ino1] = mkdir(# entries0, md2, # links(ino1, dirs), m1, entries0), root[q + str '] = dnode(md1, ∅), root.dir?, root[q].dir?, dom of < ino0, abs(root, ROOT_INO)(dirs), afs-cons(dirs, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs, files), of-cons(of, dirs, files), pw(user, md2), px(user, md2), px-parent(user, q, ROOT_INO, dirs, files), str ∈ entries0, ino1 ∈ dirs, entries0[str] ∈ dirs, psynced → asynced0, links(ino1, dirs) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ [fail#(; ; err); asynced := ?; files0 := files; vfs_put_inode#(mkinode(entries0[str], md0, true, dirs[entries0[str]].nlink, dirs[entries0[str]].nsubdirs, 0), of, asynced; err, dirs, files)] ( of-cons(of, dirs, files) ∧ afs-cons(dirs, files) → ⟨posix_rmdir#(q + str ', user, abs-files(files0), abs-of(of); err1, root; psynced)⟩ (abs-files(files0) = abs-files(files) ∧ abs(root, ROOT_INO)(dirs) ∧ (psynced → asynced) ∧ err = err1))

Rule argument

Additional information

Used simplifier rules: make-abs-fold-work: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n, md, n0, n1, entries) → # entries = n;%dir-empty: ⊦ ispath(p, ino, m, dirs, files) ∧ px-parent(user, p, ino, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) ∧ m ∈ dirs ∧ dirs[m] = mkdir(m0, md, n0, n1, entries) ∧ str ∈ entries ∧ entries[str] ∈ dirs → (dirs[entries[str]].entries = ∅ ↔ tree[p + str '].entries = ∅);%dir-entry: ⊦ ispath(p, ino, m, dirs, files) ∧ px-parent(user, p, ino, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) ∧ m ∈ dirs ∧ dirs[m] = mkdir(m0, md, n0, n1, entries) ∧ str ∈ entries → (entries[str] ∈ dirs ↔ tree[p + str '].dir?);% ⊦ dnode(md, sdt).dir?; (s)% ⊦ dnode(md, sdt).entries = sdt; (s)%⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)%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