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Proof step 21 from proof refine-rename-dir-negdentry (previous step)

# links(entries1[str], cache_dirs) = m1, # links(m, cache_dirs) = m12, # links(m3, cache_dirs) = m9, old_child_inode.ino = old_dent.ino, asynced1 ↔ cache_sync, cache_dirs[m3] = mkdir(m8, md4, # links(m3, cache_dirs), m10, entries1), cache_dirs[m] = mkdir(m11, md2, # links(m, cache_dirs), m13, entries), cache_dirs[entries1[str]] = mkdir(m0, md3, # links(entries1[str], cache_dirs), m2, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, new_child_inode = new_child_inode0, new_dent = negdentry(str0), new_parent_inode = mkinode(m, md, true, # links(m, cache_dirs), m13, m11), old_child_inode = mkinode(entries1[str], md1, true, # links(entries1[str], cache_dirs), m2, m0), old_dent = mkdentry(str, entries1[str]), old_parent_inode = mkinode(m3, md0, true, # links(m3, cache_dirs), m10, m8), pcache0 = cache_pcache, 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), valid-dentry(old_parent_inode.ino, old_dent, cache_dirs, cache_files), valid-dir-inode(old_parent_inode, cache_dirs, cache_files), valid-dir-inode(new_parent_inode, cache_dirs, cache_files), valid-dir-inode(old_child_inode, cache_dirs, cache_files), valid-negdentry(new_parent_inode.ino, new_dent, cache_dirs, cache_files), m ∈ cache_dirs, m3 ∈ cache_dirs, entries1[str] ∈ cache_dirs, str ∈ entries1, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), new_parent_inode.ino = old_parent_inode.ino → new_parent_inode.meta = old_parent_inode.meta, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ [cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)] ( icache-cons(cache_icache, cache_dirs, cache_files) ∧ dcache-cons(cache_dcache, cache_dirs, cache_files) ∧ pcache-cons(pcache0, cache_dirs, cache_files) ∧ afs-cons(cache_dirs, cache_files) ∧ synced(cache_dirs, cache_icache, pcache0) ∧ (asynced1 → synced(cache_files, cache_icache, pcache0)) → ⟨old_dent0 := mkdentry(str, entries1[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m, md, true, m12, m13, m11); old_parent_inode0 := mkinode(m3, md0, true, m9, m10, m8); old_child_inode0 := mkinode(entries1[str], md1, true, m1, m2, m0); afs_rename# (; old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode0 ∧ new_parent_inode = new_parent_inode0 ∧ old_child_inode = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced))), str0 ∈ entries

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 22:
# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries1[str], cache_dirs) = cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].nlink = m1, cache_dirs[m].nlink = m12, cache_dirs[m3].nlink = m9, entries1[str] = cache_dirs[m3].entries[str], cache_dirs[m3] = mkdir(cache_dirs[m3].size, md4, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), cache_dirs[m] = mkdir(cache_dirs[m].size, md2, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries), cache_dirs[entries1[str]] = mkdir(cache_dirs[entries1[str]].size, md3, cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].nsubdirs, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m0 = cache_dirs[entries1[str]].size, m10 = cache_dirs[m3].nsubdirs, m11 = cache_dirs[m].size, m13 = cache_dirs[m].nsubdirs, m2 = cache_dirs[entries1[str]].nsubdirs, m8 = cache_dirs[m3].size, new_child_inode = new_child_inode0, new_dent = negdentry(str0), new_parent_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), old_child_inode = mkinode(entries1[str], md1, true, cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].nsubdirs, cache_dirs[entries1[str]].size), old_dent = mkdentry(str, entries1[str]), old_parent_inode = mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].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[m3].entries, m ∈ cache_dirs, m3 ∈ cache_dirs, entries1[str] ∈ cache_dirs, str ∈ entries1, ¬ str0 ∈ cache_dirs[m].entries, ¬ str0 ∈ entries, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ [cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)] ( icache-cons(cache_icache, cache_dirs, cache_files) ∧ dcache-cons(cache_dcache, cache_dirs, cache_files) ∧ pcache-cons(cache_pcache, cache_dirs, cache_files) ∧ afs-cons(cache_dirs, cache_files) ∧ synced(cache_dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨old_dent0 := mkdentry(str, entries1[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m, md, true, m12, m13, m11); old_parent_inode0 := mkinode(m3, md0, true, m9, m10, m8); old_child_inode0 := mkinode(entries1[str], md1, true, m1, m2, m0); afs_rename# (; old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode0 ∧ new_parent_inode = new_parent_inode0 ∧ old_child_inode = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced)))

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% ⊦ mkinode(m, md, boolvar, m0, m1, m2).meta = md; (s)% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ mkdentry(str, m).name = str; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (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)% ⊦ 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)% ⊦ negdentry(str).negdentry?; (s)% ⊦ negdentry(str).name = str; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (s)%⊦ valid-negdentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ ¬ dent.name ∈ dirs[ino].entries ∧ dent.negdentry?; (s)% ⊦ mkdentry(str, m).ino = m; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)