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Proof step 37 from proof refine-link (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, entries[str] = cache_dirs[ino].entries[str], cache_files[entries[str]] = mkfile(m3, md6, # links(entries[str], cache_dirs), content1), entries0[str] = entries[str], cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m5, md1, # links(entries[str], cache_dirs), content), cache_dirs[ino1] = mkdir(m0, md3, # links(ino1, cache_dirs), m2, entries0), cache_dirs[ino0] = mkdir(m4, md2, # links(ino0, cache_dirs), m10, entries), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, (cache_dirs[m].entries:= cache_dirs[m].entries[str0, entries[str]]).size:= (cache_dirs[m].size + 1)], 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[ino].entries, ino ∈ cache_dirs, str ∈ entries, ino0 ∈ cache_dirs, ino1 ∈ cache_dirs, str ∈ entries0, m ∈ cache_dirs, entries[str] ∈ cache_files, ¬ str0 ∈ entries1, ¬ str0 ∈ cache_dirs[m].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(ino, cache_dirs) ≠ ∅ ∨ ino = ROOT_INO, links(ino0, cache_dirs) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino1, cache_dirs) ≠ ∅ ∨ ino1 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_inc_nlink#(mkdentry(str, entries0[str]).ino; dirs, cache_files); new_dent := mkdentry(str0, mkdentry(str, entries0[str]).ino); afs_inode_sync#(mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, cache_files); afs_inode_sync#(mkinode(entries[str], md0, false, # links(entries[str], cache_dirs), 0, m6); dirs, cache_files); p_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, cache_files[m].meta, false, cache_files[m].nlink, 0, cache_files[m].size) ); c_inode := getinode(mkdentry(str, entries0[str]).ino, dirs, cache_files); err := ESUCCESS; asynced := ?; 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, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); p_inode1 := mkinode(entries0[str], md0, false, # links(entries[str], cache_dirs), 0, m6); afs_link#(mkdentry(str, entries0[str]); p_inode0, p_inode1, old_dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced) ∧ 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 38:
# links(m, cache_dirs) = m7, cache_dirs[ino0] = mkdir(m4, md2, # links(ino0, cache_dirs), m10, entries), cache_dirs[ino1] = mkdir(m0, md3, # links(ino1, cache_dirs), m2, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m5, md1, # links(entries[str], cache_dirs), content), cache_dirs[m] = mkdir(m1, md5, m7, m8, entries1), entries0[str] = entries[str], cache_files[entries[str]] = mkfile(m3, md6, # links(entries[str], cache_dirs), content1), entries[str] = cache_dirs[ino].entries[str], cache_dirs[m] = mkdir(m1, md4, m7, m8, entries2), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, (cache_dirs[m].entries:= entries2[str0, entries[str]]).size:= (m1 + 1)], 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), entries[str] ∈ cache_files, m ∈ cache_dirs, str ∈ entries0, ino1 ∈ cache_dirs, ino0 ∈ cache_dirs, str ∈ entries, ino ∈ cache_dirs, str ∈ cache_dirs[ino].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(ino1, cache_dirs) ≠ ∅ ∨ ino1 = ROOT_INO, links(ino0, cache_dirs) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino, cache_dirs) ≠ ∅ ∨ ino = ROOT_INO ⊦ ⟪afs_inc_nlink#(mkdentry(str, entries0[str]).ino; dirs, cache_files); new_dent := mkdentry(str0, mkdentry(str, entries0[str]).ino); afs_inode_sync#(mkinode(m, md, true, m7, m8, m1); dirs, cache_files); afs_inode_sync#(mkinode(entries[str], md0, false, # links(entries[str], cache_dirs), 0, m6); dirs, cache_files); p_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, cache_files[m].meta, false, cache_files[m].nlink, 0, cache_files[m].size) ); c_inode := getinode(mkdentry(str, entries0[str]).ino, dirs, cache_files); err := ESUCCESS; asynced := ?; 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, # links(m, cache_dirs), m8, m1); p_inode1 := mkinode(entries0[str], md0, false, # links(entries[str], cache_dirs), 0, m6); afs_link#(mkdentry(str, entries0[str]); p_inode0, p_inode1, old_dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ new_dent = old_dent0 ∧ err = err0), str0 ∈ entries1, str0 ∈ entries2

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], m1, md4, m7, m8, entries2]

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], m1, md4, m7, m8, entries2