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Proof step 20 from proof crash-neutral-evict (previous step)

# links(m, dirs) = 0, dirs[m] = mkdir(m0, md, 0, m2, ∅), afs-cons(dirs, files), m ∈ dirs ⊦ dirs \ orphans(dirs, files) = (dirs \ (orphans(dirs, files) -- m)) -- m, m = ROOT_INO

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ dirs = dirs0 ↔ (∀ ino. (ino ∈ dirs ↔ ino ∈ dirs0) ∧ (ino ∈ dirs → dirs[ino] = dirs0[ino]))
Proof step 22:
# links(m, dirs) = 0, dirs[m] = mkdir(m0, md, 0, m2, ∅), afs-cons(dirs, files), m ∈ dirs ⊦ m = ROOT_INO, ∀ ino. (ino ∈ dirs \ orphans(dirs, files) ↔ ino ∈ (dirs \ (orphans(dirs, files) -- m)) -- m) ∧ (ino ∈ dirs \ orphans(dirs, files) → (dirs \ orphans(dirs, files))[ino] = ((dirs \ (orphans(dirs, files) -- m)) -- m)[ino])

Rule argument

spec: heap, inst: dir-heap name: ext seq: ⊦ dirs = dirs0 ↔ (∀ ino. (ino ∈ dirs ↔ ino ∈ dirs0) ∧ (ino ∈ dirs → dirs[ino] = dirs0[ino])) subst: [dirs0, dirs] → [(dirs \ (orphans(dirs, files) -- m)) -- m, dirs \ orphans(dirs, files)] flags: <all paths>

Additional information

Used terms: (dirs \ (orphans(dirs, files) -- m)) -- m, dirs \ orphans(dirs, files)