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Proof step 12 from proof abs-unfold (previous step)

dirs[ino] = mkdir(# entries, md, # links(ino, dirs), m1, entries), ((abs|(tree, ROOT_INO, p, ino) * abs(tree[p], ino)))(dirs), tree[p].dir?, afs-cons(dirs, files), fids-cons(tree, abs-files(files), ofh), ispath(p, ROOT_INO, ino, dirs, files), ino ∈ dirs, p ∈ tree ⊦ abs(tree0, ROOT_INO)(dirs0), ∃ sdt. ((abs|(tree, ROOT_INO, p, ino) * ino => mkdir(# entries, md, # links(ino, dirs), m1, entries) * abs(sdt, entries)))(dirs) ∧ tree[p] = dnode(md, sdt)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ tree.dir? → abs(tree, ino) = _∃(λ ad. ino => ad * abs(tree.entries, ad.entries) ∧ ⌜ad.meta = tree.meta⌝ ∧ ⌜ad.size = # ad.entries⌝ )
Proof step 14:
dirs[ino] = mkdir(# entries, md, # links(ino, dirs), m1, entries), ((abs|(tree, ROOT_INO, p, ino) * _∃(λ ad. ino => ad * abs(tree[p].entries, ad.entries) ∧ ⌜ad.meta = tree[p].meta⌝ ∧ ⌜ad.size = # ad.entries⌝ )) )(dirs), tree[p].dir?, afs-cons(dirs, files), fids-cons(tree, abs-files(files), ofh), ispath(p, ROOT_INO, ino, dirs, files), p ∈ tree, ino ∈ dirs ⊦ abs(tree0, ROOT_INO)(dirs0), ∃ sdt. ((abs|(tree, ROOT_INO, p, ino) * ino => mkdir(# entries, md, # links(ino, dirs), m1, entries) * abs(sdt, entries)))(dirs) ∧ tree[p] = dnode(md, sdt)

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] → [ino, tree[p]] flags: <all paths>

Additional information

Used terms: ino, tree[p]