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Proof step 29 from proof afs-cons-create-ino (previous step)

(ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);mkdir(m6, md3, m7, m8, entries0)) = mkdir(m9, md4, m10, m11, entries1), (files ++ ino)[ino0] = mkfile(m4, md2, m5, content0), files[ino0] = mkfile(m1, md1, m3, content), dirs[m] = mkdir(m0, md0, # links(m, dirs), m2, entries), dirs[ino0] = mkdir(m6, md3, m7, m8, entries0), ino0 = m, afs-cons(dirs, files), ROOT_INO ∈ dirs, m ∈ dirs, ino0 ∈ dirs → m7 = # links(ino0, dirs) ∧ # links(ino0, dirs) ≤ 1 ∧ m6 = # entries0 ∧ m8 = # subdirs(ino0, dirs), ino0 ∈ dirs ∧ name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files, ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries0 = ∅, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, ino0 = ino, ino ∈ files, ino ∈ dirs, str ∈ entries, ino0 ∈ files, (ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → entries1 = ∅) ∧ ( ino0 ∈ dirs → m10 = # links(ino0, dirs) ∧ # links(ino0, dirs) ≤ 1 ∧ m9 = # entries1 ∧ m11 = # subdirs(ino0, dirs[m, mkdir(m0, md0, # links(m, dirs), m2, entries[str, ino])])) ∧ (ino0 ∈ dirs ∧ name ∈ entries1 → entries1[name] ∈ dirs ∨ entries1[name] = ino ∨ entries1[name] ∈ files)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 30:
dirs[m] = mkdir(# entries0, md3, # links(m, dirs), # subdirs(m, dirs), entries0), files[m] = mkfile(m1, md1, m3, content), (files ++ ino)[m] = mkfile(m4, md2, m5, content0), afs-cons(dirs, files), m ∈ dirs, ROOT_INO ∈ dirs, # links(m, dirs) ≤ 1, m ≠ ROOT_INO ∧ links(m, dirs) = ∅ → entries0 = ∅, name ∈ entries0 → entries0[name] ∈ dirs ∨ entries0[name] ∈ files, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, str ∈ entries0, ino ∈ dirs, ino ∈ files, (m = ROOT_INO ∨ links(m, dirs) ≠ ∅) ∧ (name ∈ entries0[str, ino] → entries0[str, ino][name] ∈ dirs ∨ entries0[str, ino][name] = ino ∨ entries0[str, ino][name] ∈ files)

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)% ⊦ ¬ str ∈ entries → # entries[str, ino] = # entries + 1; (s)% ⊦ entries[str, ino] ≠ ∅; (s)%afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base%unc: ⊦ strings ∪ strings0 = strings0 ∪ strings; from specification set-union[dir-entries]%una: ⊦ (strings ∪ strings0) ∪ strings1 = strings ∪ strings0 ∪ strings1; from specification set-union[dir-entries]% ⊦ strings ∪ ∅ = strings; (s)% ⊦ strings \ ∅ = strings; (s)%⊦ p_ino ∈ dirs ∧ dirs[p_ino] = mkdir(m, md, n0, n1, entries) → subdirs (p_ino, dirs[p_ino, mkdir(m, md, n0, n1, entries[str, ino])]) = ( subdirs(p_ino, dirs) \ (str ∈ entries ∧ entries[str] ∈ dirs ⊃ {str} ;∅)) ∪ (ino ∈ dirs ⊃ {str} ;∅); (s)% ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs → ¬ ino ∈ files; (s)%⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)