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Proof step 78 from proof refine-readmeta (previous step)

root[path] = dnode(md1, sdt), ((m => mkdir(# entries, md1, # links(m, dirs), # subdirs(m, dirs), entries) * abs(sdt, entries) * abs|(root, ROOT_INO, path, m)))(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, m ≠ ROOT_INO ∧ links(m, dirs) = ∅ → entries = ∅, 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 ∈ entries → entries[name] ∈ dirs ∨ entries[name] ∈ files ⊦ # subdirs(m, dirs) = # dnode(md1, sdt).subdirs, str ∈ entries ∧ entries[str] ∈ dirs ↔ str ∈ sdt ∧ sdt[str].dir?, m ∈ files

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 79:
str ∈ sdt ⊦ ((abs(sdt, sdi) * D))(dirs) ↔ (∃ tree, ino, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ sdi = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * D))(dirs))
Proof step 80:
root[path] = dnode(md1, sdt), 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, str ∈ sdt, # links(m, dirs) ≤ 1, psynced → asynced, m ≠ ROOT_INO ∧ links(m, dirs) = ∅ → entries = ∅, ∀ name. name ∈ entries → entries[name] ∈ dirs ∨ entries[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), ∃ tree, ino, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ entries = sdi0[str, ino] ∧ (( abs(tree, ino) * abs(sdt0, sdi0) * abs|(root, ROOT_INO, path, m) * m => mkdir(# entries, md1, # links(m, dirs), # subdirs(m, dirs), entries)) )(dirs) ⊦ # subdirs(m, dirs) = # dnode(md1, sdt).subdirs, str ∈ entries ∧ entries[str] ∈ dirs ↔ str ∈ sdt ∧ sdt[str].dir?, m ∈ files

Rule argument

spec: vfs-abs name: entries-unfold seq: str ∈ sdt ⊦ ((abs(sdt, sdi) * D))(dirs) ↔ (∃ tree, ino, sdi0, sdt0. ¬ str ∈ sdt0 ∧ sdt = sdt0[str, tree] ∧ ¬ str ∈ sdi0 ∧ sdi = sdi0[str, ino] ∧ ((abs(tree, ino) * abs(sdt0, sdi0) * D))(dirs)) subst: [str, dirs, D, sdi, sdt] → [str, dirs, abs|(root, ROOT_INO, path, m) * m => mkdir (# entries, md1, # links(m, dirs), # subdirs(m, dirs), entries), entries, sdt] flags: <all paths>

Additional information

Used simplifier rules: entries-dom1: ⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt; (forward) from specification vfs-abs%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] Used terms: str, dirs, abs|(root, ROOT_INO, path, m) * m => mkdir (# entries, md1, # links(m, dirs), # subdirs(m, dirs), entries), entries, sdt