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Proof step 25 from proof refine-link (previous step)

# links(m, cache_dirs) = m8, # links(entries0[str], cache_dirs) = m3, cache_files[entries0[str]] = mkfile(m1, md4, # links(entries0[str], cache_dirs), content0), entries0[str] = entries[str], cache_dirs[m] = mkdir(m7, md5, # links(m, cache_dirs), m11, entries1), ((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[ino0] = mkdir(m4, md2, m9, m10, entries), 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, new_dent = negdentry(str0), p_inode = mkinode(m, md, true, # links(m, cache_dirs), 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 ∈ 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, m11, m7); 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 26:
# links(m, cache_dirs) = m8, # links(entries[str], cache_dirs) = m3, cache_files[entries[str]] = mkfile(m1, md4, # links(entries[str], cache_dirs), content0), entries0[str] = entries[str], cache_dirs[m] = mkdir(m7, md5, # links(m, cache_dirs), m11, entries1), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m5, md1, # links(entries[str], cache_dirs), content), cache_dirs[ino] = mkdir(m0, md3, # links(ino, cache_dirs), m2, entries0), cache_dirs[ino0] = mkdir(m4, md2, # links(ino0, cache_dirs), m10, entries), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(entries[str], md0, false, # links(entries[str], cache_dirs), 0, m6), err = err0, new_dent = negdentry(str0), p_inode = mkinode(m, md, true, # links(m, cache_dirs), 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 ∈ entries, ino0 ∈ cache_dirs, ino ∈ cache_dirs, str ∈ entries0, m ∈ cache_dirs, entries[str] ∈ cache_files, ¬ str0 ∈ entries1, 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, m11, m7); 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)

Rule argument

Additional information

Used simplifier rules: afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base