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Proof step 157 from proof refine-readmeta (previous step)

dirs[m9] = mkdir(# entries, md, # links(m9, dirs), m4, entries), root[q + str '] = fnode(⌜m⌝ ), dirs[m1] = mkdir(m5, md0, m6, m7, entries0), files[m] = mkfile(m0, md1, # links(m, dirs), content), ((abs|(root, ROOT_INO, q, m9) * _∃(λ ad. m9 => ad * abs(root[q].entries, ad.entries) ∧ ⌜ad.meta = root[q].meta⌝ ∧ ⌜ad.size = # ad.entries⌝ )) )(dirs), root.dir?, root[q].dir?, dom of < ino6, afs-cons(dirs, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(path, ROOT_INO, m, dirs, files), ispath(q, ROOT_INO, m9, dirs, files), of-cons(of, dirs, files), pr(user, md1), px-parent(user, path, ROOT_INO, dirs, files), q ∈ root, ROOT_INO ∈ dirs, m ∈ files, str ' ∈ root[q], m9 ∈ dirs, str ∈ entries, entries[str] ∈ files, psynced → asynced ⊦ # links(m, dirs) = count(⌜m⌝ , root.fids), # links(⌜m⌝ , root) = 0, links(⌜m⌝ , root) = ∅, m = entries[str], root[path].dir?, m ∈ dirs, entries[str] ∈ dirs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 158:
files[m] = mkfile(m0, md1, # links(m, dirs), content), dirs[m1] = mkdir(m5, md0, m6, m7, entries0), root[q + str '] = fnode(⌜m⌝ ), ((m9 => mkdir(# entries, root[q].meta, # links(m9, dirs), m4, entries) * abs(root[q].entries, entries) * abs|(root, ROOT_INO, q, m9)) )(dirs), root[q].dir?, root.dir?, dom of < ino6, afs-cons(dirs, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(q, ROOT_INO, m9, dirs, files), ispath(path, ROOT_INO, m, dirs, files), of-cons(of, dirs, files), pr(user, md1), px-parent(user, path, ROOT_INO, dirs, files), str ∈ root[q].entries, entries[str] ∈ files, str ' ∈ root[q], m ∈ files, ROOT_INO ∈ dirs, q ∈ root, str ∈ entries, psynced → asynced ⊦ # links(⌜m⌝ , root) = 0, # links(m, dirs) = count(⌜m⌝ , root.fids), links(⌜m⌝ , root) = ∅, m = entries[str], root[path].dir?, entries[str] ∈ dirs, m ∈ dirs

Rule argument

Additional information

Used simplifier rules: entries-dom2: ⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi; (forward) from specification vfs-abs%entries-dom1: ⊦ str ∈ sdi ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdt; (forward) from specification vfs-abs% ⊦ 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)% ⊦ ((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)