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Proof step 81 from proof refine-readmeta (previous step)

root[path] = dnode(md1, sdt0[str, tree]), (( abs(tree, ino) * abs(sdt0, sdi) * abs|(root, ROOT_INO, path, m) * m => mkdir(# sdi + 1, md1, # links(m, dirs), # subdirs(m, dirs), sdi[str, ino])) )(dirs), root.dir?, dom of < ino6, fids-cons(root, abs-files(files), abs-of(of)), ispath(path, ROOT_INO, m, dirs, files), of-cons(of, dirs, files), pr(user, md1), px-parent(user, path, ROOT_INO, dirs, files), ROOT_INO ∈ dirs, path ∈ root, # links(m, dirs) ≤ 1, psynced → asynced, ∀ ino. (ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → dirs[ino].entries = ∅) ∧ ( ino ∈ dirs → ¬ ino ∈ files ∧ dirs[ino].nlink = # links(ino, dirs) ∧ dirs[ino].nlink ≤ 1 ∧ dirs[ino].size = # dirs[ino].entries ∧ dirs[ino].nsubdirs = # subdirs(ino, dirs)) ∧ (ino ∈ files → ¬ ino ∈ dirs ∧ files[ino].nlink = # links(ino, dirs)) ∧ (∀ name. ino ∈ dirs ∧ name ∈ dirs[ino].entries → dirs[ino].entries[name] ∈ dirs ∨ dirs[ino].entries[name] ∈ files) ∧ (∀ n. ino ∈ files ∧ n ∈ files[ino].content → # files[ino].content[n] = VFS_PAGE_SIZE), ∀ name. name ∈ sdi[str, ino] → sdi[str, ino][name] ∈ dirs ∨ sdi[str, ino][name] ∈ files ⊦ # subdirs(m, dirs) = # dnode(md1, sdt0[str, tree]).subdirs, ino ∈ dirs ↔ tree.dir?, m ∈ files, str ∈ sdi, str ∈ sdt0, m ≠ ROOT_INO ∧ links(m, dirs) = ∅

User applied rule case distinction

And obtained the following new goals

Proof step 82:
root[path] = dnode(md1, sdt0[str, tree]), (( abs(tree, ino) * abs(sdt0, sdi) * abs|(root, ROOT_INO, path, m) * m => mkdir(# sdi + 1, md1, # links(m, dirs), # subdirs(m, dirs), sdi[str, ino])) )(dirs), root.dir?, dom of < ino6, fids-cons(root, abs-files(files), abs-of(of)), ispath(path, ROOT_INO, m, dirs, files), of-cons(of, dirs, files), pr(user, md1), px-parent(user, path, ROOT_INO, dirs, files), path ∈ root, ROOT_INO ∈ dirs, ino ∈ dirs, # links(m, dirs) ≤ 1, psynced → asynced, ∀ name. name ∈ sdi[str, ino] → sdi[str, ino][name] ∈ dirs ∨ sdi[str, ino][name] ∈ files, ∀ ino. (ino ∈ dirs ∧ ino ≠ ROOT_INO ∧ links(ino, dirs) = ∅ → dirs[ino].entries = ∅) ∧ ( ino ∈ dirs → ¬ ino ∈ files ∧ dirs[ino].nlink = # links(ino, dirs) ∧ dirs[ino].nlink ≤ 1 ∧ dirs[ino].size = # dirs[ino].entries ∧ dirs[ino].nsubdirs = # subdirs(ino, dirs)) ∧ (ino ∈ files → ¬ ino ∈ dirs ∧ files[ino].nlink = # links(ino, dirs)) ∧ (∀ name. ino ∈ dirs ∧ name ∈ dirs[ino].entries → dirs[ino].entries[name] ∈ dirs ∨ dirs[ino].entries[name] ∈ files) ∧ (∀ n. ino ∈ files ∧ n ∈ files[ino].content → # files[ino].content[n] = VFS_PAGE_SIZE) ⊦ # subdirs(m, dirs) = # dnode(md1, sdt0[str, tree]).subdirs, tree.dir?, str ∈ sdt0, str ∈ sdi, m ∈ files, m ≠ ROOT_INO ∧ links(m, dirs) = ∅

Rule argument

pos: right 2

Additional information

Used simplifier rules: ⊦ ¬ ino ∈ dirs → (((abs(tree, ino) * D))(dirs) ↔ tree = fnode(⌜ino⌝ ) ∧ D(dirs)); (ws)%sep-ass: ⊦ (D * D0) * D1 = D * D0 * D1; from specification heap-sep[dir-heap]%sep-com: ⊦ D * D0 = D0 * D; from specification heap-sep[dir-heap]% ⊦ ¬ fnode(fid).dir?; (s)