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Proof step 8 from proof path-fid (previous step)

dirs[ino0] = mkdir(m, md, m0, m1, entries), tree = dnode(md0, sdt), ((abs(tree, ino0) * D))(dirs), ispath(p0, entries[str], ino1, dirs, files), str ' ∈ tree, p0 ∈ tree[str '], ino1 ∈ files, ino0 ∈ dirs, str ∈ entries, ∀ tree, ino0, D, dirs, ino1, files. ((abs(tree, ino0) * D))(dirs) ∧ ispath(p0, ino0, ino1, dirs, files) ∧ p0 ∈ tree ∧ ino1 ∈ files → tree[p0] = fnode(⌜ino1⌝ ) ∨ ino1 ∈ dirs ⊦ tree[str ' + p0] = fnode(⌜ino1⌝ ), ino1 ∈ dirs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
⊦ abs(dnode(md, sdt), ino) = _∃(λ ad. ino => ad * abs(sdt, ad.entries) ∧ ⌜ad.meta = md⌝ ∧ ⌜ad.size = # ad.entries⌝ )
Proof step 10:
⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi
Proof step 11:
⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt
Proof step 12:
((ino0 => mkdir(# entries, md, m0, m1, entries) * abs(sdt, entries) * D))(dirs), ispath(p0, entries[str], ino1, dirs, files), str ∈ entries, ino1 ∈ files, p0 ∈ sdt[str], str ∈ sdt, ∀ tree, ino0, D, dirs, ino1, files. ((abs(tree, ino0) * D))(dirs) ∧ ispath(p0, ino0, ino1, dirs, files) ∧ p0 ∈ tree ∧ ino1 ∈ files → tree[p0] = fnode(⌜ino1⌝ ) ∨ ino1 ∈ dirs ⊦ sdt[str][p0] = fnode(⌜ino1⌝ ), ino1 ∈ 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-dom2: ⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi;%entries-dom1: ⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt;% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)% ⊦ mkdir(m, md, m0, m1, entries).size = m; (s)%str ∈ sdt ⊦ dnode(md, sdt)[str '] = sdt[str]; (s)%path-in-dirs: p ≠ ε ⊦ ispath(p, n, m, dirs, files) → n ∈ dirs; (forward) from specification afs-base% ⊦ ((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)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ ¬ str ∈ sdt → sdt[str] = (∅)[str]; (s)% ⊦ str ∈ sdt → dnode(md, sdt)[str ' + p] = sdt[str][p]; (s)% ⊦ str ' ∈ dnode(md, sdt) ↔ str ∈ sdt; (s)