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Proof step 49 from proof abs-acyclic (previous step)

sdt[str] = dnode(md, sdt0), entries[str] = ROOT_INO, tree[p] = dnode(md0, sdt), tree = dnode(md1, sdt1), (( ( ( (ROOT_INO => mkdir(# entries0, md, m0, m1, entries0) * abs(sdt0, entries0)) * ino => mkdir(# entries, md0, # links(ino, dirs), m2, entries)) * abs(sdt -- str, entries -- str)) * abs|(tree, ROOT_INO, p, ino)) )(dirs), abs(tree, ROOT_INO)(dirs), afs-cons(dirs, files), fids-cons(tree, abs-files(files), ofh), ispath(p, ROOT_INO, ino, dirs, files), str ∈ entries, str ∈ sdt ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 50:
⊦ ispath(p, ino, m, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → p ∈ tree ∧ (tree[p].dir? ↔ m ∈ dirs)
Proof step 51:
⊦ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p ∈ tree ↔ ispath(p, ino, dirs, files))
Proof step 52:
dnode(md1, sdt1)[p] = dnode(md0, sdt), entries[str] = ROOT_INO, sdt[str] = dnode(md, sdt0), (( ( ( (ROOT_INO => mkdir(# entries0, md, m0, m1, entries0) * abs(sdt0, entries0)) * ino => mkdir(# entries, md0, # links(ino, dirs), m2, entries)) * abs(sdt -- str, entries -- str)) * abs|(dnode(md1, sdt1), ROOT_INO, p, ino)) )(dirs), abs(dnode(md1, sdt1), ROOT_INO)(dirs), afs-cons(dirs, files), fids-cons(dnode(md1, sdt1), abs-files(files), ofh), ispath(p, ROOT_INO, ino, dirs, files), str ∈ sdt, str ∈ entries ⊦

Rule argument

Additional information

Used simplifier rules: path-in: ⊦ ispath(p, ino, m, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → p ∈ tree ∧ (tree[p].dir? ↔ m ∈ dirs);%path-in-eq: ⊦ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p ∈ tree ↔ ispath(p, ino, dirs, files));% ⊦ dnode(md, sdt).dir?; (s)% ⊦ ispath(p, n, m, dirs, files) → ispath(p, n, dirs, files); (s)% ⊦ ((ino => ad * D))(dirs) → ino ∈ 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]