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Proof step 29 from proof abs-acyclic (previous step)

tree[p] = dnode(md0, sdt), entries[str] = ROOT_INO, (( ino => mkdir(# entries, md0, # links(ino, dirs), m2, entries) * (abs(sdt -- str, entries -- str) * abs(sdt[str], ROOT_INO)) * 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 ∈ sdt, str ∈ entries ⊦ sdt[str].dir?

User applied rule apply lemma

And obtained the following new goals

Proof step 30:
∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ((abs(tree, ino) * D))(dirs), ispath(p, ino, ino0, dirs, files) ⊦ tree[p].dir? ↔ ino0 ∈ dirs
Proof step 31:
entries[str] = ROOT_INO, tree[p] = dnode(md0, sdt), (( ino => mkdir(# entries, md0, # links(ino, dirs), m2, entries) * (abs(sdt -- str, entries -- str) * abs(sdt[str], ROOT_INO)) * 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 ⊦ sdt[str].dir?, (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) ∧ (∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files) ∧ ((abs(tree, ROOT_INO) * emp))(dirs) ∧ ispath(p + str, ROOT_INO, entries[str], dirs, files)
Proof step 38:
tree[p + str].dir? ↔ entries[str] ∈ dirs, entries[str] = ROOT_INO, tree[p] = dnode(md0, sdt), (( ino => mkdir(# entries, md0, # links(ino, dirs), m2, entries) * (abs(sdt -- str, entries -- str) * abs(sdt[str], ROOT_INO)) * 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 ⊦ sdt[str].dir?

Rule argument

spec: vfs-abs name: abs-dir-h seq: ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ((abs(tree, ino) * D))(dirs), ispath(p, ino, ino0, dirs, files) ⊦ tree[p].dir? ↔ ino0 ∈ dirs subst: [files, dirs, tree, ino, D, p, ino0] → [files, dirs, tree, ROOT_INO, emp, p + str, entries[str]]

Additional information

Used terms: files, dirs, tree, ROOT_INO, emp, p + str, entries[str]