Proof step 16 from proof afs-cons-create-ino (previous step)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 17:

  dirs[m] = mkdir(m0, md0, # links(m, dirs), m2, entries), afs-cons(dirs, files), ROOT_INO ∈ dirs, m ∈ dirs, ino0 ∈ files ∧ n ∈ files[ino0].content → # files[ino0].content[n] = VFS_PAGE_SIZE, ino0 ∈ dirs ∧ name ∈ dirs[ino0].entries → dirs[ino0].entries[name] ∈ dirs ∨ dirs[ino0].entries[name] ∈ files, ino0 ∈ files → ¬ ino0 ∈ dirs ∧ files[ino0].nlink = # links(ino0, dirs), ino0 ∈ dirs → dirs[ino0].nlink = # links(ino0, dirs) ∧ dirs[ino0].nlink ≤ 1 ∧ dirs[ino0].size = # dirs[ino0].entries ∧ dirs[ino0].nsubdirs = # subdirs(ino0, dirs), ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → dirs[ino0].entries = ∅, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ino = 0, ino ∈ files, ino ∈ dirs, str ∈ entries, ( ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries = ∅) ∧ ( ino0 ∈ dirs → (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).nlink = # links(ino0, dirs) ∧ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).nlink ≤ 1 ∧ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).size = # (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries ∧ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).nsubdirs = # subdirs(ino0, dirs[m, mkdir(m0, md0, # links(m, dirs), m2, entries[str, ino])])) ∧ ( ino0 = ino ∨ ino0 ∈ files → ¬ ino0 ∈ dirs ∧ (ino0 = ino ⊃ mkfile(0, md, 1, ∅);(files ++ ino)[ino0]).nlink = # (ino = ino0 ⊃ ∅ ++ mklink(m, str);links(ino0, dirs))) ∧ ( ino0 ∈ dirs ∧ name ∈ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries → (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries[name] ∈ dirs ∨ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries[name] = ino ∨ (ino0 = m ⊃ mkdir(m0 + 1, md0, # links(m, dirs), m2, entries[str, ino]);dirs[ino0]).entries[name] ∈ files) ∧ ( (ino0 = ino ∨ ino0 ∈ files) ∧ n ∈ (ino0 = ino ⊃ mkfile(0, md, 1, ∅);(files ++ ino)[ino0]).content → # (ino0 = ino ⊃ mkfile(0, md, 1, ∅);(files ++ ino)[ino0]).content[n] = VFS_PAGE_SIZE)

Rule argument

Additional information

Used simplifier rules: afs-cons-reduce: ⊦ afs-cons(dirs0, files0) → ( afs-cons(dirs1, files1) ↔ (∀ ino, name, n. ROOT_INO ∈ dirs0 ∧ ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dir-empty(ino, dirs0, files0)) ∧ ( ino ∈ dirs0 → ¬ ino ∈ files0 ∧ dirs0[ino].nlink = # links(ino, dirs0) ∧ dirs0[ino].nlink ≤ 1 ∧ dirs0[ino].size = # dirs0[ino].entries ∧ dirs0[ino].nsubdirs = # subdirs(ino, dirs0)) ∧ ( ino ∈ files0 → ¬ ino ∈ dirs0 ∧ files0[ino].nlink = # links(ino, dirs0)) ∧ ( ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → valid-ino(dirs0[ino].entries[name], dirs0, files0)) ∧ ( ino ∈ files0 ∧ n ∈ files0[ino].content → valid-pagesize(ino, n, files0)) → ROOT_INO ∈ dirs1 ∧ ( ino ∈ dirs1 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs1) = ∅ → dir-empty(ino, dirs1, files1)) ∧ ( ino ∈ dirs1 → ¬ ino ∈ files1 ∧ dirs1[ino].nlink = # links(ino, dirs1) ∧ dirs1[ino].nlink ≤ 1 ∧ dirs1[ino].size = # dirs1[ino].entries ∧ dirs1[ino].nsubdirs = # subdirs(ino, dirs1)) ∧ ( ino ∈ files1 → ¬ ino ∈ dirs1 ∧ files1[ino].nlink = # links(ino, dirs1)) ∧ ( ino ∈ dirs1 ∧ name ∈ dirs1[ino].entries → valid-ino(dirs1[ino].entries[name], dirs1, files1)) ∧ ( ino ∈ files1 ∧ n ∈ files1[ino].content → valid-pagesize(ino, n, files1))));%New-same: ⊦ p_ino ∈ dirs ∧ afs-cons(dirs, files) ∧ ino ≠ 0 ∧ ¬ ino ∈ files ∧ ¬ ino ∈ dirs → ¬ p_ino ∈ (files ++ ino)[ino, af];%⊦ afs-cons(dirs, files) ∧ ¬ ino ∈ dirs ∧ ¬ ino ∈ files → links(ino, dirs) = ∅; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%⊦ p_ino ∈ dirs → subdirs(ino, dirs[p_ino, mkdir(m0 + 1, md, n0, n1, entries)]) = subdirs(ino, dirs[p_ino, mkdir(m0, md, n0, n1, entries)]); (s)%⊦ ino ∈ dirs ∧ dirs[ino] = mkdir(m0, md0, n1, n2, entries) → links(ino0, dirs[ino, mkdir(m, md, n, n0, entries)]) = links(ino0, dirs); (s)%⊦ p_ino ∈ dirs → links (ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries[str, ino])]) = (ino = ino0 ⊃ links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) ++ mklink(p_ino, str) ;(str ∈ entries ∧ entries[str] = ino0 ⊃ links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) -- mklink(p_ino, str) ;links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) ) ); (s)%⊦ mkdir(m, md, m0, m1, entries).size:= m2 = mkdir(m2, md, m0, m1, entries); (s)% ⊦ mkdir(m, md, m0, m1, entries).size = m; (s)%⊦ mkdir(m, md, m0, m1, entries).entries:= entries0 = mkdir(m, md, m0, m1, entries0); (s)% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)% ⊦ files[ino, af][ino0] = (ino0 = ino ∧ ino ∈ files ⊃ af;files[ino0]); (s)%ino ≠ 0, ¬ ino ∈ files ⊦ ino0 ∈ files ++ ino ↔ ino0 = ino ∨ ino0 ∈ files; (s)% ⊦ ino ∈ files[ino0, af] ↔ ino ∈ files; (s)% ⊦ afs-cons(dirs, files) → ROOT_INO ∈ dirs; (ss)% ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (s)%⊦ valid-pagesize(ino, n, files) ↔ # files[ino].content[n] = VFS_PAGE_SIZE; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs → ¬ ino ∈ files; (s)% ⊦ dir-empty(ino, dirs, files) ↔ dirs[ino].entries = ∅; (s)% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)% ⊦ (ad.entries:= entries).size = ad.size; (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)% ⊦ negdentry(str).name = str; (s)