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Proof step 215 from proof refine-write_inode (previous step)

cache_files[m] = mkfile(m0, md0, # links(m, cache_dirs), content), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_files, cache_icache, cache_pcache), synced(cache_dirs, cache_icache, cache_pcache), m ∈ cache_files ⊦ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[m, mkfile(m2, md, # links(m, cache_dirs), content ⊕ mkcontent(m, cache_pcache))] = (cache_files[m, mkfile(m2, md, # links(m, cache_dirs), content)] ⊕ cache_pcache) ⊕ cache_icache, m ∈ cache_icache

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 216:
⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino]))
Proof step 217:
cache_files[m] = mkfile(m0, md0, # links(m, cache_dirs), content), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), synced(cache_files, cache_icache, cache_pcache), m ∈ cache_files ⊦ m ∈ cache_icache, ∀ ino. ( ino ∈ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[m, mkfile(m2, md, # links(m, cache_dirs), content ⊕ mkcontent(m, cache_pcache))] ↔ ino ∈ (cache_files[m, mkfile(m2, md, # links(m, cache_dirs), content)] ⊕ cache_pcache) ⊕ cache_icache) ∧ ( ino ∈ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[m, mkfile(m2, md, # links(m, cache_dirs), content ⊕ mkcontent(m, cache_pcache))] → ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[m, mkfile(m2, md, # links(m, cache_dirs), content ⊕ mkcontent(m, cache_pcache))][ino] = ((cache_files[m, mkfile(m2, md, # links(m, cache_dirs), content)] ⊕ cache_pcache) ⊕ cache_icache)[ino])

Rule argument

spec: heap, inst: file-heap name: ext seq: ⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino])) subst: [files, files0] → [((cache_files ⊕ cache_pcache) ⊕ cache_icache )[m, mkfile (m2, md, # links(m, cache_dirs), content ⊕ mkcontent(m, cache_pcache)) ], ( cache_files [m, mkfile(m2, md, # links(m, cache_dirs), content) ] ⊕ cache_pcache) ⊕ cache_icache] flags: <all paths>

Additional information

Used terms: ((cache_files ⊕ cache_pcache) ⊕ cache_icache )[m, mkfile (m2, md, # links(m, cache_dirs), content ⊕ mkcontent(m, cache_pcache)) ], ( cache_files[m, mkfile(m2, md, # links(m, cache_dirs), content)] ⊕ cache_pcache) ⊕ cache_icache