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Proof step 22 from proof dir-empty (previous step)

dirs[entries[str]] = mkdir(# entries0, md0, # links(entries[str], dirs), m2, entries0), ((m => mkdir(# entries, tree[p].meta, # links(m, dirs), n1, entries) * abs(tree[p].entries, entries) * abs|(tree, ino, p, m)))(dirs), tree[p].dir?, tree[p][str '].dir?, afs-cons(dirs, files), fids-cons(tree, abs-files(files), ofh), ispath(p, ino, m, dirs, files), px-parent(user, p, ino, dirs, files), str ∈ tree[p].entries, entries[str] ∈ dirs, p ∈ tree, str ∈ entries ⊦ entries0 = ∅ ↔ tree[p][str '].entries = ∅

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 23:
dirs[entries[str]] = mkdir(# entries0, md0, # links(entries[str], dirs), m2, entries0), tree[p] = dnode(md, sdt), ((m => mkdir(# entries, md, # links(m, dirs), n1, entries) * abs(sdt, entries) * abs|(tree, ino, p, m)))(dirs), tree[p][str '].dir?, tree[p].dir?, afs-cons(dirs, files), fids-cons(tree, abs-files(files), ofh), ispath(p, ino, m, dirs, files), px-parent(user, p, ino, dirs, files), str ∈ entries, p ∈ tree, entries[str] ∈ dirs, str ∈ sdt ⊦ entries0 = ∅ ↔ tree[p][str '].entries = ∅

Rule argument

spec: directory-tree name: dnode-elim seq: tree.dir? ⊦ md = tree.meta ∧ sdt = tree.entries ↔ tree = dnode(md, sdt) subst: [tree, md, sdt] → [tree[p], md, sdt]

Additional information

Used simplifier rules: tree.dir? ⊦ md = tree.meta ∧ sdt = tree.entries ↔ tree = dnode(md, sdt); (elim) Used terms: tree[p], md, sdt