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Proof step 23 from proof refine-rename-dir-negdentry (previous step)

# links(entries1[str], cache_dirs) = cache_dirs[entries1[str]].nlink, # links(m, cache_dirs) = m5, # links(m3, cache_dirs) = cache_dirs[m3].nlink, cache_dirs[m3].nlink = m9, cache_dirs[entries1[str]].nlink = m1, cache_dirs[entries1[str]] = mkdir(cache_dirs[entries1[str]].size, md3, cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].nsubdirs, entries0), cache_dirs[m] = mkdir(m4, md2, m5, m6, entries), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md4, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), entries1[str] = cache_dirs[m3].entries[str], cache_dirs[m] = mkdir(m4, md5, m5, m6, entries2), 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 = m4, m13 = m6, m2 = cache_dirs[entries1[str]].nsubdirs, m5 = m12, m8 = cache_dirs[m3].size, new_child_inode = new_child_inode0, new_dent = negdentry(str0), new_parent_inode = mkinode(m, md, true, m5, m6, m4), 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), 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), str ∈ entries1, entries1[str] ∈ cache_dirs, m3 ∈ cache_dirs, m ∈ cache_dirs, str ∈ cache_dirs[m3].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = 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))), str0 ∈ entries2, str0 ∈ entries

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 24:
# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries1[str], cache_dirs) = cache_dirs[entries1[str]].nlink, # links(m, cache_dirs) = m12, cache_dirs[entries1[str]].nlink = m1, 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(m4, md5, # links(m, cache_dirs), m6, entries2), 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, 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, # links(m, cache_dirs), m6, m4), 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 ∈ entries2, 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, m6, m4); 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: ⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)