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Proof step 111 from proof refine-create (previous step)

entries[str] ∈ dirs ↔ root[q + str '].dir?, dirs[ino1] = mkdir(# entries, md0, # links(ino1, dirs), m1, entries), fs = abs-files(files), path = q + str ', 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, md0), px(user, md0), px-parent(user, q, ROOT_INO, dirs, files), ino1 ∈ dirs, str ∈ entries, psynced → asynced, links(ino1, dirs) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ ispath(path, ROOT_INO, dirs, files) ∧ ⟨err0 := EEXISTS; if err0 = ESUCCESS then choose fid with ¬ fid ∈ fs in {root[path] := fnode(fid); fs[fid] := mkfile(md, mkbuf(0))}; psynced := ?⟩ (fs = abs-files(files) ∧ abs(root, ROOT_INO)(dirs) ∧ (psynced → asynced) ∧ EEXISTS = err0)

Heuristic dl case distinction applied rule case distinction

And obtained the following new goals

Proof step 112:
entries[str] ∈ dirs ↔ root[q + str '].dir?, dirs[ino1] = mkdir(# entries, md0, # links(ino1, dirs), m1, entries), fs = abs-files(files), path = q + str ', 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, md0), px(user, md0), px-parent(user, q, ROOT_INO, dirs, files), str ∈ entries, ino1 ∈ dirs, psynced → asynced, links(ino1, dirs) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ ⟨err0 := EEXISTS; if err0 = ESUCCESS then choose fid with ¬ fid ∈ fs in {root[path] := fnode(fid); fs[fid] := mkfile(md, mkbuf(0))}; psynced := ?⟩ (fs = abs-files(files) ∧ abs(root, ROOT_INO)(dirs) ∧ (psynced → asynced) ∧ EEXISTS = err0)

Rule argument

pos: right 1

Additional information

Used simplifier rules: ⊦ ispath(p, n, m, dirs, files) → (ispath(p + q, n, dirs, files) ↔ ispath(q, m, dirs, files)); (s)%⊦ ispath(str ', n, dirs, files) ↔ n ∈ dirs ∧ str ∈ dirs[n].entries ∧ (dirs[n].entries[str] ∈ dirs ∨ dirs[n].entries[str] ∈ files); (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)%⊦ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) ∧ str ∈ entries ∧ afs-cons(dirs, files) → (entries[str] ∈ files ↔ ¬ entries[str] ∈ dirs); (ss)