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Proof step 233 from proof refine-unlink (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries0[str], cache_dirs) = cache_files[entries0[str]].nlink, entries0[str] = cache_dirs[m].entries[str], cache_dirs[m] = mkdir(cache_dirs[m].size, md1, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m7, md2, cache_files[entries0[str]].nlink, content0), cache_files[entries0[str]] = mkfile(m0, md4, cache_files[entries0[str]].nlink, content1), 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), str ∈ cache_dirs[m].entries, m ∈ cache_dirs, str ∈ entries0, entries0[str] ∈ cache_files, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [fail#(; ; err); asynced := ?; p_inode := mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size); dent := mkdentry(str, entries0[str]); c_inode := mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; if err = ESUCCESS then let asynced = false in {icache_set#(c_inode, asynced; err, cache_icache); icache_set#(p_inode, asynced; err, cache_icache); dcache_delete#(p_inode.ino, dent.name; err, cache_dcache)}] ( icache-cons(cache_icache, cache_dirs, cache_files) ∧ dcache-cons(cache_dcache, cache_dirs, cache_files) ∧ synced(cache_dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); p_inode1 := mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); cache_dirs0 := cache_dirs; afs_unlink#(; p_inode0, p_inode1, dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = cache_dirs0 ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 234:
# links(entries0[str], cache_dirs) = m2, # links(m, cache_dirs) = cache_dirs[m].nlink, cache_files[entries0[str]] = mkfile(m0, md4, m2, content1), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m7, md2, m2, content0), cache_dirs[m] = mkdir(cache_dirs[m].size, md1, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), entries0[str] = cache_dirs[m].entries[str], cache_files[entries0[str]] = mkfile(m1, md3, m2, 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_dirs, cache_icache, cache_pcache), entries0[str] ∈ cache_files, str ∈ entries0, m ∈ cache_dirs, str ∈ cache_dirs[m].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [fail#(; ; err); asynced := ?; p_inode := mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size); dent := mkdentry(str, entries0[str]); c_inode := mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; if err = ESUCCESS then let asynced = false in {icache_set#(c_inode, asynced; err, cache_icache); icache_set#(p_inode, asynced; err, cache_icache); dcache_delete#(p_inode.ino, dent.name; err, cache_dcache)}] ( icache-cons(cache_icache, cache_dirs, cache_files) ∧ dcache-cons(cache_dcache, cache_dirs, cache_files) ∧ synced(cache_dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); p_inode1 := mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); cache_dirs0 := cache_dirs; afs_unlink#(; p_inode0, p_inode1, dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = cache_dirs0 ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0))

Rule argument

spec: abstract-file name: elim seq: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content) subst: [af, m, md, m0, content] → [cache_files[entries0[str]], m1, md3, m2, content]

Additional information

Used simplifier rules: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content); (elim) Used terms: cache_files[entries0[str]], m1, md3, m2, content