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Proof step 102 from proof walk-prepost-h (previous step)

dirs[entries0[str]] = mkdir(m, md, # links(entries0[str], dirs), m2, entries), dirs[ino0] = mkdir(m0, md0, # links(ino0, dirs), m3, entries0), afs-cons(dirs, files), ispath(q, ROOT_INO, ino0, dirs, files), px(user, md0), px(user, parent(q), ROOT_INO, dirs, files), ino0 ∈ dirs, str ∈ entries0, entries0[str] ∈ dirs, links(ino0, dirs) ≠ ∅ ∨ ino0 = ROOT_INO, ∀ p1, err, p, p_ino, ino, dirs, files, q, user, asynced. # p1 < #(str + p2) → 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))) ⊦ p2 = ε, ispath(q + str ', ROOT_INO, entries0[str], dirs, files), px(user, md), ∃ p'. p' ⊑ str ' + parent(p2) ∧ ispath(p', ino0, dirs, files) ∧ ¬ px(user, q + p', ROOT_INO, dirs, files)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: afs-base name: path-trans seq: ⊦ ispath(p, n0, n1, dirs, files) ∧ ispath(q, n1, n2, dirs, files) → ispath(p + q, n0, n2, dirs, files) subst: [n1, files, dirs, n2, n0, q, p] → [ino0, files, dirs, entries0[str], ROOT_INO, str ', q] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ispath(str ', n, m, dirs, files) ↔ n ∈ dirs ∧ str ∈ dirs[n].entries ∧ (m ∈ dirs ∨ m ∈ files) ∧ dirs[n].entries[str] = m; (s)% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s) Used terms: ino0, files, dirs, entries0[str], ROOT_INO, str ', q