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Proof step 19 from proof refine-link (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
# links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[m].nlink = m8, entries0[str] = cache_dirs[ino0].entries[str], cache_dirs[m] = mkdir(cache_dirs[m].size, md2, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m5, md1, cache_files[entries0[str]].nlink, content), cache_dirs[ino] = mkdir(m0, md3, # links(ino, cache_dirs), m2, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(entries0[str], md0, false, cache_files[entries0[str]].nlink, 0, m6), err = err0, m4 = cache_dirs[m].size, m7 = cache_files[entries0[str]].nlink, 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 ⊕ 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, ¬ str0 ∈ cache_dirs[m].entries, ¬ str0 ∈ entries, 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, m7, 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% ⊦ valid-file-ino(ino, dirs, files) ↔ ino ∈ files; (s)%⊦ valid-file-inode(inode, dirs, files) ↔ valid-file-ino(inode.ino, dirs, files) ∧ inode.nlink = files[inode.ino].nlink ∧ inode.nsubdirs = 0 ∧ ¬ inode.directory; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).directory ↔ boolvar; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nsubdirs = m1; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).size = m2; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nlink = m0; (s)% ⊦ valid-dir-ino(ino, dirs, files) ↔ ino ∈ dirs; (s)%⊦ valid-dir-inode(inode, dirs, files) ↔ valid-dir-ino(inode.ino, dirs, files) ∧ inode.nlink = dirs[inode.ino].nlink ∧ inode.size = dirs[inode.ino].size ∧ inode.nsubdirs = dirs[inode.ino].nsubdirs ∧ inode.directory; (s)%⊦ p_ino ∈ dirs ∧ dirs[p_ino] = mkdir(m, md, n, n0, entries) ∧ str ∈ entries → links(entries[str], dirs) ≠ ∅; (s)% ⊦ negdentry(str).negdentry?; (s)% ⊦ negdentry(str).name = str; (s)%⊦ valid-negdentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ ¬ dent.name ∈ dirs[ino].entries ∧ dent.negdentry?; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ mkdentry(str, m).name = str; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ mkdentry(str, m).ino = m; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (s)%⊦ valid-dentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ valid-ino(dent.ino, dirs, files) ∧ dent.name ∈ dirs[ino].entries ∧ dent.ino = dirs[ino].entries[dent.name] ∧ dent.dentry?; (s)%⊦ valid-dentry(dent, dirs, files) ↔ (∃ ino. valid-dentry(ino, dent, dirs, files)); (s)%⊦ pre-link(p_inode, c_inode, old_dent, new_dent, dirs, files) ↔ valid-dentry(old_dent, dirs, files) ∧ valid-negdentry(p_inode.ino, new_dent, dirs, files) ∧ (links(old_dent.ino, dirs) ≠ ∅ ∨ old_dent.ino = ROOT_INO) ∧ valid-dir-inode(p_inode, dirs, files) ∧ valid-file-inode(c_inode, dirs, files) ∧ c_inode.ino = old_dent.ino; (s)