Proof step 28 from proof write-inode-inv (previous step)

User applied rule insert rewrite lemma

And obtained the following new goals

• Proof step 29:

  ⊦ 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))))

• Proof step 30:

  files1 -- ino = files0 -- ino, files0[ino] = mkfile(m, md, m0, content0), files1[ino] = mkfile(m, md, # links(ino, dirs), content), afs-cons(dirs, files1), write-inv(0, n1, buf, n, content0, content), ino ∈ files0, ino ∈ files1, n ≤ # buf, n1 ≤ # buf, n ≤ n1 ⊦ ∀ ino0, name, n0. ROOT_INO ∈ dirs ∧ (ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → dir-empty(ino0, dirs, files1)) ∧ ( ino0 ∈ dirs → ¬ ino0 ∈ files1 ∧ dirs[ino0].nlink = # links(ino0, dirs) ∧ dirs[ino0].nlink ≤ 1 ∧ dirs[ino0].size = # dirs[ino0].entries ∧ dirs[ino0].nsubdirs = # subdirs(ino0, dirs)) ∧ (ino0 ∈ files1 → ¬ ino0 ∈ dirs ∧ files1[ino0].nlink = # links(ino0, dirs)) ∧ (ino0 ∈ dirs ∧ name ∈ dirs[ino0].entries → valid-ino(dirs[ino0].entries[name], dirs, files1)) ∧ (ino0 ∈ files1 ∧ n0 ∈ files1[ino0].content → valid-pagesize(ino0, n0, files1)) → ROOT_INO ∈ dirs ∧ ( ino0 ∈ dirs ∧ ino0 ≠ ROOT_INO ∧ links(ino0, dirs) = ∅ → dir-empty(ino0, dirs, files1[ino, mkfile(m + n, md, # links(ino, dirs), content)])) ∧ ( ino0 ∈ dirs → ¬ ino0 ∈ files1[ino, mkfile(m + n, md, # links(ino, dirs), content)] ∧ dirs[ino0].nlink = # links(ino0, dirs) ∧ dirs[ino0].nlink ≤ 1 ∧ dirs[ino0].size = # dirs[ino0].entries ∧ dirs[ino0].nsubdirs = # subdirs(ino0, dirs)) ∧ ( ino0 ∈ files1[ino, mkfile(m + n, md, # links(ino, dirs), content)] → ¬ ino0 ∈ dirs ∧ files1[ino, mkfile(m + n, md, # links(ino, dirs), content)][ino0].nlink = # links(ino0, dirs)) ∧ ( ino0 ∈ dirs ∧ name ∈ dirs[ino0].entries → valid-ino(dirs[ino0].entries[name], dirs, files1[ino, mkfile(m + n, md, # links(ino, dirs), content)])) ∧ ( ino0 ∈ files1[ino, mkfile(m + n, md, # links(ino, dirs), content)] ∧ n0 ∈ files1[ino, mkfile(m + n, md, # links(ino, dirs), content)][ino0].content → valid-pagesize(ino0, n0, files1[ino, mkfile(m + n, md, # links(ino, dirs), content)]))

Rule argument

Additional information

Used terms: dirs, files1, files1[ino, mkfile(m + n, md, # links(ino, dirs), content)], dirs