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Proof step 568 from proof refine-rmdir (previous step)

# links(entries[str], dirs) = dirs[entries[str]].nlink, # entries = dirs[ino1].size, # links(ino1, dirs) = dirs[ino1].nlink, dirs[entries[str]].entries = ∅, 0 = dirs[entries[str]].size, entries[str] = dirs[ino1].entries[str], dirs[entries[str]] = mkdir(0, md0, dirs[entries[str]].nlink, dirs[entries[str]].nsubdirs, ∅), dirs[ino1] = mkdir(# entries, md, dirs[ino1].nlink, dirs[ino1].nsubdirs, entries), 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, md), px(user, md), px-parent(user, q, ROOT_INO, dirs, files), str ∈ dirs[ino1].entries, str ∈ entries, ino1 ∈ dirs, entries[str] ∈ dirs, 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 elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 569:
# 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))

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [dirs[ino1], m, md2, m0, m1, entries0]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: dirs[ino1], m, md2, m0, m1, entries0