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Proof step 47 from proof afs-cons-mkdir-ino (previous step)

dirs[ino0] = mkdir(# entries1, md2, # links(ino0, dirs), # subdirs(ino0, dirs), entries1), files[ino0] = mkfile(m1, md1, m3, content), dirs[m] = mkdir(m0, md0, # links(m, dirs), m2, entries), afs-cons(dirs, files), ROOT_INO ∈ dirs, m ∈ dirs, ino0 ∈ dirs, # links(ino0, dirs) ≤ 1, name ∈ entries1 → entries1[name] ∈ dirs ∨ entries1[name] ∈ files, ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries1 = ∅, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, ino0 = m, str ∈ entries, ino ∈ files, ino ∈ dirs, # subdirs(ino0, dirs) = # subdirs(ino0, dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino) ∧ (name ∈ entries1 → entries1[name] = ino ∨ entries1[name] ∈ dirs ∨ entries1[name] ∈ files)

User applied rule case distinction

And obtained the following new goals

Proof step 48:
dirs[m] = mkdir(m0, md0, # links(m, dirs), m2, entries), files[ino0] = mkfile(m1, md1, m3, content), dirs[ino0] = mkdir(# entries1, md2, # links(ino0, dirs), # subdirs(ino0, dirs), entries1), afs-cons(dirs, files), ino0 ∈ dirs, m ∈ dirs, ROOT_INO ∈ dirs, # links(ino0, dirs) ≤ 1, ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries1 = ∅, name ∈ entries1 → entries1[name] ∈ dirs ∨ entries1[name] ∈ files, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ # subdirs(ino0, dirs) = # subdirs(ino0, dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino), ino = 0, ino0 = m, ino ∈ dirs, ino ∈ files, str ∈ entries

Rule argument

pos: right 6

Additional information

Used simplifier rules: ⊦ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) ∧ str ∈ entries ∧ afs-cons(dirs, files) → (entries[str] ∈ files ↔ ¬ entries[str] ∈ dirs); (ss)