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Proof step 20 from proof refine-rename-dir-dentry (previous step)

# links(entries[str], cache_dirs) = m16, # links(m, cache_dirs) = m9, cache_dirs[m] = mkdir(m8, md4, # links(m, cache_dirs), m10, entries), cache_dirs[entries[str]] = mkdir(m15, md3, # links(entries[str], cache_dirs), m17, ∅), cache_dirs[cache_dirs[m3].entries[str0]] = mkdir(m0, md5, m1, m2, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m11 = cache_dirs[m3].entries[str0], m12 = m1, m13 = m2, m14 = m0, m4 = cache_dirs[m3].nlink, m5 = cache_dirs[m3].nsubdirs, m6 = cache_dirs[m3].size, new_child_inode = mkinode(entries[str], md1, true, # links(entries[str], cache_dirs), m17, m15), new_dent = mkdentry(str, entries[str]), new_parent_inode = mkinode(m, md, true, # links(m, cache_dirs), m10, m8), old_child_inode = mkinode(cache_dirs[m3].entries[str0], md2, true, m1, m2, m0), old_dent = mkdentry(str0, cache_dirs[m3].entries[str0]), 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), m3 ∈ cache_dirs, m ∈ cache_dirs, cache_dirs[m3].entries[str0] ∈ cache_dirs, entries[str] ∈ cache_dirs, str0 ∈ cache_dirs[m3].entries, str ∈ 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)⟫ ⟨new_dent0 := mkdentry(str, entries[str]); old_dent0 := mkdentry(str0, m11); new_parent_inode0 := mkinode(m, md, true, m9, m10, m8); old_parent_inode0 := mkinode(m3, md0, true, m4, m5, m6); new_child_inode0 := mkinode(entries[str], md1, true, m16, m17, m15); old_child_inode0 := mkinode(m11, md2, true, m12, m13, m14); 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)), entries[str] = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
# links(cache_dirs[m3].entries[str0], cache_dirs) = m1, # links(entries[str], cache_dirs) = m16, # links(m, cache_dirs) = m9, cache_dirs[m] = mkdir(m8, md4, # links(m, cache_dirs), m10, entries), cache_dirs[entries[str]] = mkdir(m15, md3, # links(entries[str], cache_dirs), m17, ∅), cache_dirs[cache_dirs[m3].entries[str0]] = mkdir(m0, md5, # links(cache_dirs[m3].entries[str0], cache_dirs), m2, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m11 = cache_dirs[m3].entries[str0], m4 = cache_dirs[m3].nlink, m5 = cache_dirs[m3].nsubdirs, m6 = cache_dirs[m3].size, new_child_inode = mkinode(entries[str], md1, true, # links(entries[str], cache_dirs), m17, m15), new_dent = mkdentry(str, entries[str]), new_parent_inode = mkinode(m, md, true, # links(m, cache_dirs), m10, m8), old_child_inode = mkinode(cache_dirs[m3].entries[str0], md2, true, # links(cache_dirs[m3].entries[str0], cache_dirs), m2, m0), old_dent = mkdentry(str0, cache_dirs[m3].entries[str0]), old_parent_inode = mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size), entries[str] ≠ ROOT_INO, 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), m3 ∈ cache_dirs, m ∈ cache_dirs, cache_dirs[m3].entries[str0] ∈ cache_dirs, entries[str] ∈ cache_dirs, str0 ∈ cache_dirs[m3].entries, str ∈ 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)⟫ ⟨new_dent0 := mkdentry(str, entries[str]); old_dent0 := mkdentry(str0, m11); new_parent_inode0 := mkinode(m, md, true, m9, m10, m8); old_parent_inode0 := mkinode(m3, md0, true, m4, m5, m6); new_child_inode0 := mkinode(entries[str], md1, true, m16, m17, m15); old_child_inode0 := mkinode(m11, md2, 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