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Proof step 51 from proof inv-link (previous step)

(ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);mkdir(m11, md5, m12, m13, entries2)) = mkdir(m14, md6, m15, m16, entries3), dirs0[ino] = mkdir(m11, md5, m12, m13, entries2), dirs0[ino0] = mkdir(m2, md3, # links(ino0, dirs0), m6, entries0), files0[entries[str]] = mkfile(m1, md4, # links(entries[str], dirs0), content0), dirs0[m] = mkdir(m0, md1, # links(m, dirs0), m5, entries1), entries0[str] = entries[str], dirs0[ino1] = mkdir(m4, md2, # links(ino1, dirs0), m8, entries), files0[ino] = mkfile(m3, md, m10, content1), ino = m, afs-cons(dirs0, files0), ROOT_INO ∈ dirs0, entries[str] ∈ files0, m ∈ dirs0, str ∈ entries0, ino0 ∈ dirs0, ino1 ∈ dirs0, str ∈ entries, ino ∈ dirs0 ∧ name ∈ entries2 → entries2[name] ∈ dirs0 ∨ entries2[name] ∈ files0, ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m10 = # links(ino, dirs0), ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries2 = ∅, ino ∈ dirs0 → m12 = # links(ino, dirs0) ∧ # links(ino, dirs0) ≤ 1 ∧ m11 = # entries2 ∧ m13 = # subdirs(ino, dirs0), links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(ino0, dirs0) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ ino = entries[str], str0 ∈ entries1, (ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries3 = ∅) ∧ ( ino ∈ dirs0 → m15 = # links(ino, dirs0) ∧ # links(ino, dirs0) ≤ 1 ∧ m14 = # entries3 ∧ m16 = # subdirs(ino, dirs0[m, mkdir(m0, md1, # links(m, dirs0), m5, entries1[str0, entries[str]])])) ∧ (ino ∈ dirs0 ∧ name ∈ entries3 → entries3[name] ∈ dirs0 ∨ entries3[name] ∈ files0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 52:
files0[entries[str]] = mkfile(m1, md4, # links(entries[str], dirs0), content0), dirs0[ino0] = mkdir(m2, md3, # links(ino0, dirs0), m6, entries0), dirs0[m] = mkdir(# entries1, md5, # links(m, dirs0), # subdirs(m, dirs0), entries1), files0[m] = mkfile(m3, md, m10, content1), dirs0[ino1] = mkdir(m4, md2, # links(ino1, dirs0), m8, entries), entries0[str] = entries[str], afs-cons(dirs0, files0), str ∈ entries, ino1 ∈ dirs0, ino0 ∈ dirs0, str ∈ entries0, m ∈ dirs0, entries[str] ∈ files0, ROOT_INO ∈ dirs0, # links(m, dirs0) ≤ 1, m ≠ ROOT_INO ∧ links(m, dirs0) = ∅ → entries1 = ∅, name ∈ entries1 → entries1[name] ∈ dirs0 ∨ entries1[name] ∈ files0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO, links(ino0, dirs0) ≠ ∅ ∨ ino0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ m = entries[str], str0 ∈ entries1, (m = ROOT_INO ∨ links(m, dirs0) ≠ ∅) ∧ (name ∈ entries1[str0, entries[str]] → entries1[str0, entries[str]][name] ∈ dirs0 ∨ entries1[str0, entries[str]][name] ∈ files0)

Rule argument

Additional information

Used simplifier rules: 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)%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% ⊦ {str} = ∅ ++ str; (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)