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Proof step 32 from proof recovery (previous step)

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), cache_sync0 → synced(cache_files0, cache_icache0, cache_pcache0) ⊦ cache_dirs0 \ orphans(cache_dirs0, (cache_files0 ⊕ cache_pcache0) ⊕ cache_icache0) = cache_dirs0 \ orphans(cache_dirs0, cache_files0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 33:
⊦ dirs = dirs0 ↔ (∀ ino. (ino ∈ dirs ↔ ino ∈ dirs0) ∧ (ino ∈ dirs → dirs[ino] = dirs0[ino]))
Proof step 34:
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), cache_sync0 → synced(cache_files0, cache_icache0, cache_pcache0) ⊦ ∀ ino. ( ino ∈ cache_dirs0 \ orphans(cache_dirs0, (cache_files0 ⊕ cache_pcache0) ⊕ cache_icache0) ↔ ino ∈ cache_dirs0 \ orphans(cache_dirs0, cache_files0)) ∧ ( ino ∈ cache_dirs0 \ orphans(cache_dirs0, (cache_files0 ⊕ cache_pcache0) ⊕ cache_icache0) → (cache_dirs0 \ orphans(cache_dirs0, (cache_files0 ⊕ cache_pcache0) ⊕ cache_icache0))[ino] = (cache_dirs0 \ orphans(cache_dirs0, cache_files0))[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] → [cache_dirs0 \ orphans(cache_dirs0, cache_files0), cache_dirs0 \ orphans (cache_dirs0, (cache_files0 ⊕ cache_pcache0) ⊕ cache_icache0)] flags: <all paths>

Additional information

Used terms: cache_dirs0 \ orphans(cache_dirs0, cache_files0), cache_dirs0 \ orphans(cache_dirs0, (cache_files0 ⊕ cache_pcache0) ⊕ cache_icache0)