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Proof step 67 from proof refine-writemeta (previous step)

dirs0[m] = mkdir(# entries, md0, # links(m, dirs0), m2, entries), root[path] = dnode(md0, sdt), ((m => mkdir(# entries, md, # links(m, dirs0), m2, entries) * abs|(root, ROOT_INO, path, m) * abs(sdt, entries)))(dirs), root.dir?, dom of < ino0, afs-cons(dirs0, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(path, ROOT_INO, m, dirs0, files), of-cons(of, dirs0, files), pw(user, md0), px-parent(user, path, ROOT_INO, dirs0, files), m ∈ dirs0, path ∈ root, dirs ≡ dirs0[m, mkdir(# entries, md, # links(m, dirs0), m2, entries)], psynced0 → asynced0 ⊦ abs(root[path, dnode(md, sdt)], ROOT_INO)(dirs)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 68:
⊦ p ∈ tree → abs(tree, ino0) = _∃(λ ino1. abs|(tree, ino0, p, ino1) * abs(tree[p], ino1))
Proof step 69:
root[path] = dnode(md0, sdt), dirs0[m] = mkdir(# entries, md0, # links(m, dirs0), m2, entries), ((m => mkdir(# entries, md, # links(m, dirs0), m2, entries) * abs|(root, ROOT_INO, path, m) * abs(sdt, entries)))(dirs), root.dir?, dom of < ino0, afs-cons(dirs0, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(path, ROOT_INO, m, dirs0, files), of-cons(of, dirs0, files), pw(user, md0), px-parent(user, path, ROOT_INO, dirs0, files), path ∈ root, m ∈ dirs0, path ∈ root[path, dnode(md, sdt)], dirs ≡ dirs0[m, mkdir(# entries, md, # links(m, dirs0), m2, entries)], psynced0 → asynced0 ⊦ (_∃(λ ino1. abs|(root[path, dnode(md, sdt)], ROOT_INO, path, ino1) * abs(root[path, dnode(md, sdt)][path], ino1)))(dirs)

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[path, dnode(md, sdt)]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ p ∈ tree → p ∈ tree[p, tree0]; (s) Used terms: path, ROOT_INO, root[path, dnode(md, sdt)]