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Proof step 39 from proof abs-tree (previous step)

tree1 = dnode(md0, sdt0), ((abs(tree1, ino) * D1))(dirs), (((ino => mkdir(# entries, md, m0, m1, entries) * abs(sdt, entries)) * D0))(dirs), ∀ tree1, ino, D1, dirs, tree0, D0, files. ((abs(tree1, ino) * D1))(dirs) ∧ ((abs(tree0, ino) * D0))(dirs) ∧ (∀ ino. ⌜ino⌝ ∈ tree1.fids → ino ∈ files) ∧ (∀ ino. ⌜ino⌝ ∈ tree0.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (p0 ∈ tree0 ↔ p0 ∈ tree1) ∧ (p0 ∈ tree0 → (tree0[p0].dir? ↔ tree1[p0].dir?) ∧ (tree0[p0].dir? ⊃ tree0[p0].meta = tree1[p0].meta;tree0[p0].fid = tree1[p0].fid)), ∀ ino. ⌜ino⌝ ∈ tree1.fids → ino ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino0. ⌜ino0⌝ ∈ sdt.fids → ino0 ∈ files ⊦ (str ∈ sdt ∧ p0 ∈ sdt[str] ↔ str ' ∈ tree1 ∧ p0 ∈ tree1[str ']) ∧ ( str ∈ sdt ∧ p0 ∈ sdt[str] → (sdt[str][p0].dir? ↔ tree1[str ' + p0].dir?) ∧ (sdt[str][p0].dir? ⊃ sdt[str][p0].meta = tree1[str ' + p0].meta;sdt[str][p0].fid = tree1[str ' + p0].fid))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 40:
⊦ abs(dnode(md, sdt), ino) = _∃(λ ad. ino => ad * abs(sdt, ad.entries) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ )
Proof step 41:
(((ino => mkdir(# entries, md, m0, m1, entries) * abs(sdt, entries)) * D0))(dirs), ((ino => mkdir(# entries, md, m0, m1, entries) * abs(sdt0, entries) * D1))(dirs), ∀ ino0. ⌜ino0⌝ ∈ sdt0.fids → ino0 ∈ files, ∀ tree1, ino, D1, dirs, tree0, D0, files. ((abs(tree1, ino) * D1))(dirs) ∧ ((abs(tree0, ino) * D0))(dirs) ∧ (∀ ino. ⌜ino⌝ ∈ tree1.fids → ino ∈ files) ∧ (∀ ino. ⌜ino⌝ ∈ tree0.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (p0 ∈ tree0 ↔ p0 ∈ tree1) ∧ (p0 ∈ tree0 → (tree0[p0].dir? ↔ tree1[p0].dir?) ∧ (tree0[p0].dir? ⊃ tree0[p0].meta = tree1[p0].meta;tree0[p0].fid = tree1[p0].fid)), ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino0. ⌜ino0⌝ ∈ sdt.fids → ino0 ∈ files ⊦ (str ∈ sdt ∧ p0 ∈ sdt[str] ↔ str ∈ sdt0 ∧ p0 ∈ sdt0[str]) ∧ ( str ∈ sdt ∧ p0 ∈ sdt[str] → (sdt[str][p0].dir? ↔ dnode(md, sdt0)[str ' + p0].dir?) ∧ (sdt[str][p0].dir? ⊃ sdt[str][p0].meta = dnode(md, sdt0)[str ' + p0].meta;sdt[str][p0].fid = dnode(md, sdt0)[str ' + p0].fid))

Rule argument

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⌝ );% ⊦ mkdir(m, md, m0, m1, entries).meta = md; (s)% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)% ⊦ mkdir(m, md, m0, m1, entries).size = m; (s)%eq-objs: ⊦ ((ino => ad * D))(dirs) ∧ ((ino => ad0 * D0))(dirs) → ad = ad0; (forward) from specification maplet[dir-heap]%str ∈ sdt ⊦ dnode(md, sdt)[str '] = sdt[str]; (s)% ⊦ str ' ∈ dnode(md, sdt) ↔ str ∈ sdt; (s)% ⊦ dnode(md, sdt).fids = sdt.fids; (s)% ⊦ ⌜bv⌝ (dirs) ↔ bv; (s)% ⊦ ((D ∧ D0))(dirs) ↔ D(dirs) ∧ D0(dirs); (s)%sep-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]%and-com: ⊦ (D ∧ D0) = (D0 ∧ D); from specification heap-sep[dir-heap]%and-ass: ⊦ ((D ∧ D0) ∧ D1) = (D ∧ D0 ∧ D1); from specification heap-sep[dir-heap]% ⊦ (⌜bv⌝ ∧ D) * D0 = (⌜bv⌝ ∧ D * D0); (s)% ⊦ ((_∃ adD * D))(dirs) ↔ (∃ ad. ((adD(ad) * D))(dirs)); (s)