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Proof step 384 from proof refine-unlink (previous step)

# links(entries0[str], dirs0) = m + 1, # entries0 = m0 + 1, files0[entries0[str]] = mkfile(m1, md1, m + 1, content0), root0[q] = dnode(md0, sdt), dirs0[ino1] = mkdir(m0 + 1, md0, # links(ino1, dirs0), m3, entries0), dirs = dirs1, err0 = ESUCCESS, files = files0[entries0[str], mkfile(m1, md1, m, content0)], fs = abs-files(files0), ((ino1 => mkdir(m0, md0, # links(ino1, dirs0), m3, entries0 -- str) * abs|(root0, ROOT_INO, q, ino1) * abs(sdt, entries0)))(dirs1), root0.dir?, (root0 -- (q + str ')).dir?, dom of < ino0, afs-cons(dirs0, files0), afs-cons(dirs1, files1), fids-cons(root0, abs-files(files0), abs-of(of)), fids-cons(root0 -- (q + str '), fs0, abs-of(of)), ispath(q, ROOT_INO, ino1, dirs0, files0), of-cons(of, dirs0, files0), of-cons(of, dirs1, files1), pw(user, md0), px(user, md0), px-parent(user, q, ROOT_INO, dirs0, files0), str ∈ entries0, q ∈ root0, entries0[str] ∈ files0, ino1 ∈ dirs0, str ∈ sdt, dirs1 ≡ dirs0[ino1, mkdir(m0, md0, # links(ino1, dirs0), m3, entries0 -- str)], psynced0 → asynced0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ entries0 = ∅, root0[q + str '].dir?, abs(root0 -- (q + str '), ROOT_INO)(dirs1), entries0[str] ∈ dirs0, entries0[str] ∈ dirs1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 385:
⊦ str ∈ sdt → abs(sdt, sdi) = (⌜str ∈ sdi⌝ ∧ abs(sdt -- str, sdi -- str) * abs(sdt[str], sdi[str]))
Proof step 386:
# entries0 = m0 + 1, # links(entries0[str], dirs0) = m + 1, dirs0[ino1] = mkdir(m0 + 1, md0, # links(ino1, dirs0), m3, entries0), root0[q] = dnode(md0, sdt), files0[entries0[str]] = mkfile(m1, md1, m + 1, content0), dirs = dirs1, err0 = ESUCCESS, files = files0[entries0[str], mkfile(m1, md1, m, content0)], fs = abs-files(files0), (( ino1 => mkdir(m0, md0, # links(ino1, dirs0), m3, entries0 -- str) * abs|(root0, ROOT_INO, q, ino1) * (⌜str ∈ entries0⌝ ∧ abs(sdt -- str, entries0 -- str) * abs(sdt[str], entries0[str]))) )(dirs1), (root0 -- (q + str ')).dir?, root0.dir?, dom of < ino0, afs-cons(dirs1, files1), afs-cons(dirs0, files0), fids-cons(root0 -- (q + str '), fs0, abs-of(of)), fids-cons(root0, abs-files(files0), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs0, files0), of-cons(of, dirs1, files1), of-cons(of, dirs0, files0), pw(user, md0), px(user, md0), px-parent(user, q, ROOT_INO, dirs0, files0), ino1 ∈ dirs0, entries0[str] ∈ files0, q ∈ root0, str ∈ entries0, str ∈ sdt, dirs1 ≡ dirs0[ino1, mkdir(m0, md0, # links(ino1, dirs0), m3, entries0 -- str)], psynced0 → asynced0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ entries0 = ∅, root0[q + str '].dir?, abs(root0 -- (q + str '), ROOT_INO)(dirs1), entries0[str] ∈ dirs1, entries0[str] ∈ dirs0

Rule argument

spec: vfs-abs name: entries-recdel seq: ⊦ str ∈ sdt → abs(sdt, sdi) = ( ⌜str ∈ sdi⌝ ∧ abs(sdt -- str, sdi -- str) * abs(sdt[str], sdi[str])) subst: [str, sdi, sdt] → [str, entries0, sdt] flags: <all paths>

Additional information

Used terms: str, entries0, sdt