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Proof step 37 from proof walk-prepost-h (previous step)

dirs[ino] = mkdir(m, md, # links(ino, dirs), m1, entries), afs-cons(dirs, files), ispath(q, ROOT_INO, ino, dirs, files), px(user, q, ROOT_INO, dirs, files), ino ∈ dirs, q ≠ ε → px(user, parent(q), ROOT_INO, dirs, files), ∀ p1, err, p, p_ino, ino, dirs, files, q, user, asynced. # p1 < # p0 → err = ESUCCESS ∧ p = p1 ∧ p_ino = ino ∧ afs-cons(dirs, files) ∧ ispath(q, ROOT_INO, ino, dirs, files) ∧ ino ∈ dirs ∧ (q ≠ ε → px(user, parent(q), ROOT_INO, dirs, files)) → [while p ≠ ε ∧ err = ESUCCESS do {choose inode with true in vfs_may_lookup#(ino, user, dirs, files, asynced; inode, err); if err = ESUCCESS then let dent = negdentry(p.head) in {afs_lookup#(ino, dirs, files, asynced; dent, err); if err = ESUCCESS then {ino := dent.ino; p := p.tail}}}] ( ⌊err⌋ ∨ err = ESUCCESS ∧ ispath(p1, p_ino, ino, dirs, files) ∧ px-parent(user, q + p1, ROOT_INO, dirs, files) ∨ err = ENOENT ∧ ¬ ispath(p1, p_ino, dirs, files) ∨ err = EACCESS ∧ p1 ≠ ε ∧ (∃ p'. p' ⊑ parent(p1) ∧ ispath(p', p_ino, dirs, files) ∧ ¬ px(user, q + p', ROOT_INO, dirs, files))) ⊦ p0 = ε, px(user, md)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: px-last seq: ⊦ px(user, parent(p), ino, dirs, files) → ( px(user, p, ino, dirs, files) ↔ px(user, meta(p, ino, dirs, files))) subst: [files, dirs, ino, p, user] → [files, dirs, ROOT_INO, q, user] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ispath(p, n, m, dirs, files) → meta(p, n, dirs, files) = meta(ε, m, dirs, files); (s)% ⊦ ino ∈ dirs → meta(ε, ino, dirs, files) = dirs[ino].meta; (s)% ⊦ mkdir(m, md, m0, m1, entries).meta = md; (s)% ⊦ ispath(ε, n, m, dirs, files) ↔ valid-ino(n, dirs, files) ∧ m = n; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ px(user, ε, ino, dirs, files) ↔ px(user, meta(ε, ino, dirs, files)); (s) Used terms: files, dirs, ROOT_INO, q, user