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Proof step 37 from proof afs-cons-mkdir-ino (previous step)

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

User applied rule case distinction

And obtained the following new goals

Proof step 38:
dirs[m] = mkdir(# entries, md0, # links(m, dirs), # subdirs(m, dirs), entries), files[m] = mkfile(m1, md1, m3, content), afs-cons(dirs, files), ROOT_INO ∈ dirs, m ∈ dirs, # links(m, dirs) ≤ 1, name ∈ entries → entries[name] ∈ dirs ∨ entries[name] ∈ files, m ≠ ROOT_INO ∧ links(m, dirs) = ∅ → entries = ∅, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ # subdirs(m, dirs) + 1 = # subdirs(m, dirs[m, mkdir(# entries + 1, md0, # links(m, dirs), # subdirs(m, dirs) + 1, entries[str, ino])] ++ ino), ino = 0, str ∈ entries, ino ∈ files, ino ∈ dirs

Rule argument

pos: right 5

Additional information

Used simplifier rules: ⊦ str ∈ entries[str0, ino] ↔ str = str0 ∨ str ∈ entries; (ws)% ⊦ str = str0 → entries[str0, ino][str] = ino; (s)%⊦ str ∈ entries ∧ ¬ str0 ∈ entries → entries[str0, ino][str] = entries[str]; (s)%⊦ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) ∧ str ∈ entries ∧ afs-cons(dirs, files) → (entries[str] ∈ files ↔ ¬ entries[str] ∈ dirs); (ss)%⊦ afs-cons(dirs, files) → ( links(ino, dirs) = ∅ ↔ (ino ∈ dirs → dirs[ino].nlink = 0) ∧ (ino ∈ files → files[ino].nlink = 0)); (ws)% ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs → ¬ ino ∈ files; (s)%size-zero-01: ⊦ ls ≠ ∅ → # ls ≠ 0; (forward) from specification set-basic[alink-set]% ⊦ mkdir(m, md, m0, m1, entries).nlink = m0; (s)