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Proof step 38 from proof afs-cons-mkdir-ino (previous step)

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

User applied rule cut formula

And obtained the following new goals

Proof step 39:
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 ⊦ # 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, subdirs(m, dirs[m, mkdir(# entries + 1, md0, # links(m, dirs), # subdirs(m, dirs) + 1, entries[str, ino])] ++ ino) = subdirs(m, dirs) ++ str, ino ∈ dirs, ino ∈ files, str ∈ entries

Rule argument

fma: subdirs (m, dirs [m, mkdir (# entries + 1, md0, # links(m, dirs), # subdirs(m, dirs) + 1, entries[str, ino]) ] ++ ino) = subdirs(m, dirs) ++ str

Additional information

Used simplifier rules: ⊦ # strings + 1 = #(strings ++ str) ↔ ¬ str ∈ strings; (s)%⊦ str ∈ subdirs(ino, dirs) ↔ ino ∈ dirs ∧ str ∈ dirs[ino].entries ∧ dirs[ino].entries[str] ∈ dirs; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)% ⊦ ¬ str ∈ entries → entries[str] = (∅)[str]; (s) Used formulas: subdirs (m, dirs [m, mkdir (# entries + 1, md0, # links(m, dirs), # subdirs(m, dirs) + 1, entries[str, ino]) ] ++ ino) = subdirs(m, dirs) ++ str