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Proof step 14 from proof abs-dir-h (previous step)

dirs[ino] = mkdir(m, md, m0, m1, entries), tree = dnode(md0, sdt), ((abs(tree, ino) * D))(dirs), ispath(p0, entries[str], ino0, dirs, files), ino ∈ dirs, str ∈ entries, ∀ tree, ino, D, dirs, ino0, files. ((abs(tree, ino) * D))(dirs) ∧ ispath(p0, ino, ino0, dirs, files) ∧ (∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (tree[p0].dir? ↔ ino0 ∈ dirs), ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs ⊦ tree[str ' + p0].dir? ↔ ino0 ∈ dirs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
⊦ abs(dnode(md, sdt), ino) = _∃(λ ad. ino => ad * abs(sdt, ad.entries) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ )
Proof step 16:
⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt
Proof step 17:
⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi
Proof step 18:
((ino => mkdir(# entries, md, m0, m1, entries) * abs(sdt, entries) * D))(dirs), ispath(p0, entries[str], ino0, dirs, files), str ∈ sdt, str ∈ entries, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ tree, ino, D, dirs, ino0, files. ((abs(tree, ino) * D))(dirs) ∧ ispath(p0, ino, ino0, dirs, files) ∧ (∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (tree[p0].dir? ↔ ino0 ∈ dirs), ∀ ino1. ⌜ino1⌝ ∈ sdt.fids → ino1 ∈ files ⊦ sdt[str][p0].dir? ↔ ino0 ∈ dirs

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⌝ );%entries-dom1: ⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt;%entries-dom2: ⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi;%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ str ∈ sdt → dnode(md, sdt)[str ' + p] = sdt[str][p]; (s)% ⊦ 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)% ⊦ dnode(md, sdt).fids = sdt.fids; (s)% ⊦ ((ino => ad * D))(dirs) → ino ∈ dirs; (s)% ⊦ ((ino => ad * D))(dirs) → (dirs[ino] = ad0 ↔ ad = ad0); (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)