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Proof step 58 from proof refine-readmeta (previous step)

dirs[m] = mkdir(# entries, md1, # links(m, dirs), m2, entries), root[path] = dnode(md, sdt), root.dir?, dom of < ino6, abs(root, ROOT_INO)(dirs), afs-cons(dirs, files), 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), m ∈ dirs, psynced → asynced ⊦ md1 = md ∧ m2 = # dnode(md, sdt).subdirs ∧ # entries = # sdt

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 59:
⊦ p ∈ tree → abs(tree, ino0) = _∃(λ ino1. abs|(tree, ino0, p, ino1) * abs(tree[p], ino1))
Proof step 60:
⊦ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p ∈ tree ↔ ispath(p, ino, dirs, files))
Proof step 61:
root[path] = dnode(md, sdt), dirs[m] = mkdir(# entries, md1, # links(m, dirs), m2, entries), root.dir?, dom of < ino6, (_∃(λ ino1. abs|(root, ROOT_INO, path, ino1) * abs(root[path], ino1)))(dirs), afs-cons(dirs, files), 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), m ∈ dirs, path ∈ root, psynced → asynced ⊦ md1 = md ∧ m2 = # dnode(md, sdt).subdirs ∧ # entries = # sdt

Rule argument

spec: vfs-abs name: path-unfold seq: ⊦ p ∈ tree → abs(tree, ino0) = _∃(λ ino1. abs|(tree, ino0, p, ino1) * abs(tree[p], ino1)) subst: [p, ino0, tree] → [path, ROOT_INO, root] flags: <all paths>

Additional information

Used simplifier rules: path-in-eq: ⊦ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p ∈ tree ↔ ispath(p, ino, dirs, files));% ⊦ ispath(p, n, m, dirs, files) → ispath(p, n, dirs, files); (s) Used terms: path, ROOT_INO, root