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Proof step 22 from proof refine-link (previous step)

# links(entries0[str], cache_dirs) = m3, # links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[m].nlink = m8, cache_dirs[ino] = mkdir(m0, md3, # links(ino, cache_dirs), m2, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m5, md1, # links(entries0[str], cache_dirs), content), cache_dirs[m] = mkdir(cache_dirs[m].size, md2, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries), entries0[str] = cache_dirs[ino0].entries[str], cache_files[entries0[str]] = mkfile(m1, md4, # links(entries0[str], cache_dirs), content0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6), err = err0, m4 = cache_dirs[m].size, m9 = cache_dirs[m].nsubdirs, new_dent = negdentry(str0), p_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), 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), entries0[str] ∈ cache_files, m ∈ cache_dirs, str ∈ entries0, ino ∈ cache_dirs, ino0 ∈ cache_dirs, str ∈ cache_dirs[ino0].entries, ¬ str0 ∈ cache_dirs[m].entries, ¬ str0 ∈ entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(ino, cache_dirs) ≠ ∅ ∨ ino = ROOT_INO, links(ino0, cache_dirs) ≠ ∅ ∨ ino0 = ROOT_INO ⊦ ⟪afs_link#(mkdentry(str, entries0[str]); p_inode, c_inode, new_dent, err, cache_dirs, cache_files; asynced0); if err = ESUCCESS then let asynced = false in {icache_set#(p_inode, asynced; err, cache_icache); icache_set#(c_inode, asynced; err, cache_icache); dcache_set#(p_inode.ino, new_dent.name, new_dent; err, cache_dcache)}⟫ ⟨old_dent0 := negdentry(str0); p_inode0 := mkinode(m, md, true, m8, m9, m4); p_inode1 := mkinode(entries0[str], md0, false, m3, 0, m6); afs_link#(mkdentry(str, entries0[str]); p_inode0, p_inode1, old_dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced0) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ new_dent = old_dent0 ∧ err = err0)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 23:
# links(m, cache_dirs) = m10, # links(entries0[str], cache_dirs) = m3, cache_files[entries0[str]] = mkfile(m1, md4, # links(entries0[str], cache_dirs), content0), entries0[str] = cache_dirs[ino0].entries[str], cache_dirs[m] = mkdir(m7, md2, m10, m11, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m5, md1, # links(entries0[str], cache_dirs), content), cache_dirs[ino] = mkdir(m0, md3, # links(ino, cache_dirs), m2, entries0), cache_dirs[m] = mkdir(m7, md5, m10, m11, entries1), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6), err = err0, m10 = m8, m4 = m7, m9 = m11, new_dent = negdentry(str0), p_inode = mkinode(m, md, true, m10, m11, m7), 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), str ∈ cache_dirs[ino0].entries, ino0 ∈ cache_dirs, ino ∈ cache_dirs, str ∈ entries0, m ∈ cache_dirs, entries0[str] ∈ cache_files, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(ino0, cache_dirs) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino, cache_dirs) ≠ ∅ ∨ ino = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_link#(mkdentry(str, entries0[str]); p_inode, c_inode, new_dent, err, cache_dirs, cache_files; asynced0); if err = ESUCCESS then let asynced = false in {icache_set#(p_inode, asynced; err, cache_icache); icache_set#(c_inode, asynced; err, cache_icache); dcache_set#(p_inode.ino, new_dent.name, new_dent; err, cache_dcache)}⟫ ⟨old_dent0 := negdentry(str0); p_inode0 := mkinode(m, md, true, m8, m9, m4); p_inode1 := mkinode(entries0[str], md0, false, m3, 0, m6); afs_link#(mkdentry(str, entries0[str]); p_inode0, p_inode1, old_dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced0) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ new_dent = old_dent0 ∧ err = err0), str0 ∈ entries1, str0 ∈ entries

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [cache_dirs[m], m7, md5, m10, m11, entries1]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: cache_dirs[m], m7, md5, m10, m11, entries1