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Proof step 48 from proof afs-cons-mkdir-ino (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 49:
∀ str. str ∈ strings ↔ str ∈ strings0 ⊦ # strings = # strings0 ↔ true
Proof step 50:
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 ⊦ # subdirs(ino0, dirs) = # subdirs(ino0, dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino), ino = 0, ino0 = m, str ∈ entries, ino ∈ files, ino ∈ dirs, ∀ str0. str0 ∈ subdirs(ino0, dirs) ↔ str0 ∈ subdirs(ino0, dirs[m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino])] ++ ino)

Rule argument

spec: set-union, inst: dir-entries name: size-eq-ext seq: ∀ str. str ∈ strings ↔ str ∈ strings0 ⊦ # strings = # strings0 ↔ true subst: [strings0, strings] → [subdirs (ino0, dirs [m, mkdir (m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino]) ] ++ ino), subdirs(ino0, dirs)] flags: <all paths>

Additional information

Used terms: subdirs (ino0, dirs [m, mkdir(m0 + 1, md0, # links(m, dirs), m2 + 1, entries[str, ino]) ] ++ ino), subdirs(ino0, dirs)