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

(((ino => mkdir(# entries, md, m0, m1, entries) * abs(sdt, entries)) * D0))(dirs), ((abs(tree1, ino) * D1))(dirs), ∀ ino0. ⌜ino0⌝ ∈ sdt.fids → ino0 ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree1.fids → ino ∈ files ⊦ tree1.dir? ∧ md = tree1.meta

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: tree1

Additional information

Used simplifier rules: abs-dnode: ⊦ abs(dnode(md, sdt), ino) = _∃ (λ ad. ino => ad * abs(sdt, ad.entries) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ );%directory-tree, entries-tree generated by ∅, dnode, fnode, [ ];;% ⊦ ((_∃ adD * D))(dirs) ↔ (∃ ad. ((adD(ad) * D))(dirs)); (s)% ⊦ (⌜bv⌝ ∧ D) * D0 = (⌜bv⌝ ∧ D * D0); (s)%and-ass: ⊦ ((D ∧ D0) ∧ D1) = (D ∧ D0 ∧ D1); from specification heap-sep[dir-heap]%and-com: ⊦ (D ∧ D0) = (D0 ∧ D); from specification heap-sep[dir-heap]%sep-ass: ⊦ (D * D0) * D1 = D * D0 * D1; from specification heap-sep[dir-heap]%sep-com: ⊦ D * D0 = D0 * D; from specification heap-sep[dir-heap]% ⊦ ((D ∧ D0))(dirs) ↔ D(dirs) ∧ D0(dirs); (s)% ⊦ ⌜bv⌝ (dirs) ↔ bv; (s)% ⊦ dnode(md, sdt).fids = sdt.fids; (s)% ⊦ dnode(md, sdt).dir?; (s)% ⊦ dnode(md, sdt).meta = md; (s)%eq-objs: ⊦ ((ino => ad * D))(dirs) ∧ ((ino => ad0 * D0))(dirs) → ad = ad0; (forward) from specification maplet[dir-heap]% ⊦ mkdir(m, md, m0, m1, entries).size = m; (s)% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)% ⊦ mkdir(m, md, m0, m1, entries).meta = md; (s)