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Proof step 96 from proof refine-readmeta (previous step)

root[path] = dnode(md1, sdt0[str, fnode(⌜m0⌝ )]), (((abs(sdt0, sdi) * abs|(root, ROOT_INO, path, m)) * m => mkdir(# sdi + 1, md1, # links(m, dirs), # subdirs(m, dirs), sdi[str, m0])) )(dirs), root.dir?, dom of < ino6, 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, m0 ∈ dirs, # links(m, dirs) ≤ 1, psynced → asynced, ∀ name. name ∈ sdi[str, m0] → sdi[str, m0][name] ∈ dirs ∨ sdi[str, m0][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), ∀ fid. fid ∈ abs-files(files) ↔ fid ∈ root.fids ∨ fid ∈ abs-of(of).fids ⊦ # subdirs(m, dirs) = # dnode(md1, sdt0[str, fnode(⌜m0⌝ )]).subdirs, root[path + str '] = fnode(⌜m0⌝ ), is-open(m0, of), m0 ∈ files, ⌜m0⌝ ∈ root.fids, str ∈ sdt0, str ∈ sdi, m ∈ files, m ≠ ROOT_INO ∧ links(m, dirs) = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: directory-tree name: split-at seq: ⊦ p ∈ tree → tree[p + q] = tree[p][q] subst: [q, p, tree] → [str ', path, root] flags: <all paths>

Additional information

Used simplifier rules: ⊦ str ∈ sdt[str0, tree] ↔ str = str0 ∨ str ∈ sdt; (ws)%str ∈ sdt ⊦ dnode(md, sdt)[str '] = sdt[str]; (s)% ⊦ sdt[str, tree][str] = tree; (s)% ⊦ fnode(fid) = fnode(fid0) ↔ fid = fid0; (s) Used terms: str ', path, root