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Proof step 19 from proof dir-empty (previous step)

dirs[entries[str]] = mkdir(# entries0, md0, # links(entries[str], dirs), m2, entries0), dirs[m] = mkdir(# entries, md, # links(m, dirs), n1, entries), ((abs|(tree, ino, p, m) * abs(tree[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), m ∈ dirs, str ∈ entries, entries[str] ∈ dirs, p ∈ tree ⊦ entries0 = ∅ ↔ tree[p][str '].entries = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ tree.dir? → abs(tree, ino) = _∃(λ ad. ino => ad * abs(tree.entries, ad.entries) ∧ ⌜ad.meta = tree.meta⌝ ∧ ⌜ad.size = # ad.entries⌝ )
Proof step 21:
dirs[m] = mkdir(# entries, md, # links(m, dirs), n1, entries), dirs[entries[str]] = mkdir(# entries0, md0, # links(entries[str], dirs), m2, entries0), ((abs|(tree, ino, p, m) * _∃(λ ad. m => ad * abs(tree[p].entries, ad.entries) ∧ ⌜ad.meta = tree[p].meta⌝ ∧ ⌜ad.size = # ad.entries⌝ )))(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), p ∈ tree, entries[str] ∈ dirs, str ∈ entries, m ∈ dirs ⊦ entries0 = ∅ ↔ tree[p][str '].entries = ∅

Rule argument

spec: vfs-abs name: abs-is-dnode seq: ⊦ tree.dir? → abs(tree, ino) = _∃ (λ ad. ino => ad * abs(tree.entries, ad.entries) ∧ ⌜ad.meta = tree.meta⌝ ∧ ⌜ad.size = # ad.entries⌝ ) subst: [ino, tree] → [m, tree[p]] flags: <all paths>

Additional information

Used terms: m, tree[p]