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Proof step 44 from proof recovery (previous step)

afs-cons(cache_dirs0, (cache_files0 ⊕ cache_pcache0) ⊕ cache_icache0), afs-cons(cache_dirs0, cache_files0), dcache-cons(cache_dcache0, cache_dirs0, cache_files0), icache-cons(cache_icache0, cache_dirs0, cache_files0), pcache-cons(cache_pcache0, cache_dirs0, cache_files0), synced(cache_dirs0, cache_icache0, cache_pcache0), ino ∈ cache_dirs0, cache_sync0 → synced(cache_files0, cache_icache0, cache_pcache0) ⊦ cache_dirs0[ino] = (cache_dirs0 \ orphans(cache_dirs0, cache_files0))[ino], links(ino, cache_dirs0) = ∅ ∧ ino ≠ ROOT_INO

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ ino ∈ dirs ∧ ¬ ino ∈ ns → (dirs \ ns)[ino] = dirs[ino]
Proof step 46:
afs-cons(cache_dirs0, cache_files0), afs-cons(cache_dirs0, (cache_files0 ⊕ cache_pcache0) ⊕ cache_icache0), dcache-cons(cache_dcache0, cache_dirs0, cache_files0), icache-cons(cache_icache0, cache_dirs0, cache_files0), pcache-cons(cache_pcache0, cache_dirs0, cache_files0), synced(cache_dirs0, cache_icache0, cache_pcache0), ino ∈ cache_dirs0, cache_sync0 → synced(cache_files0, cache_icache0, cache_pcache0) ⊦ cache_dirs0[ino] = (cache_dirs0 \ orphans(cache_dirs0, cache_files0))[ino], links(ino, cache_dirs0) = ∅ ∧ ino ≠ ROOT_INO, ino ∈ cache_dirs0 ∧ ¬ ino ∈ orphans(cache_dirs0, cache_files0)

Rule argument

spec: heap, inst: dir-heap name: at-diffs seq: ⊦ ino ∈ dirs ∧ ¬ ino ∈ ns → (dirs \ ns)[ino] = dirs[ino] subst: [ino, ns, dirs] → [ino, orphans(cache_dirs0, cache_files0), cache_dirs0] flags: <all paths>

Additional information

Used terms: ino, orphans(cache_dirs0, cache_files0), cache_dirs0