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Proof step 81 from proof refine-writemeta (previous step)

root[path] = dnode(md0, sdt), dirs0[m] = mkdir(# entries, md0, # links(m, dirs0), m2, entries), ((m => mkdir(# entries, md, # links(m, dirs0), m2, entries) * abs|(root, ROOT_INO, path, m) * abs(sdt, entries)))(dirs), root.dir?, dom of < ino0, afs-cons(dirs0, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(path, ROOT_INO, m, dirs0, files), of-cons(of, dirs0, files), pw(user, md0), px(user, parent(path), ROOT_INO, dirs0, files), path ∈ root, m ∈ dirs0, ROOT_INO ∈ dirs0, dirs ≡ dirs0[m, mkdir(# entries, md, # links(m, dirs0), m2, entries)], psynced0 → asynced0 ⊦ path = ε, (((m => mkdir(# entries, md, # links(m, dirs0), m2, entries) * abs(sdt, entries)) * abs|(root[path, dnode(md, sdt)], ROOT_INO, path, m)) )(dirs), parent(path) ∈ root ∧ root[parent(path)].dir?

User applied rule insert elim lemma

And closed the goal!

Rule argument

spec: path name: parent-elim seq: p ≠ ε ⊦ q = parent(p) ∧ str = p.last ↔ p = q + str ' subst: [p, q, str] → [path, q, str]

Additional information

Used simplifier rules: ⊦ p + str ' ≠ ε; (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ p + q ∈ tree ↔ p ∈ tree ∧ q ∈ tree[p]; (s)% ⊦ ¬ tree.dir? → (p ∈ tree ↔ p = ε); (s)% ⊦ str ' ≠ ε; (s)%p ≠ ε ⊦ q = parent(p) ∧ str = p.last ↔ p = q + str '; (elim) Used terms: path, q, str