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

# 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))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 22:
# links(m, cache_dirs) = m9, # links(entries[str], cache_dirs) = m16, # links(entries1[str0], cache_dirs) = m1, cache_dirs[entries1[str0]] = mkdir(m0, md5, # links(entries1[str0], cache_dirs), m2, entries0), cache_dirs[entries[str]] = mkdir(m15, md3, # links(entries[str], cache_dirs), m17, ∅), cache_dirs[m] = mkdir(m8, md4, # links(m, cache_dirs), m10, entries), cache_dirs[m3] = mkdir(m7, md6, m12, m13, entries1), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m11 = entries1[str0], m4 = m12, m5 = m13, m6 = m7, 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(entries1[str0], md2, true, # links(entries1[str0], cache_dirs), m2, m0), old_dent = mkdentry(str0, entries1[str0]), old_parent_inode = mkinode(m3, md0, true, m12, m13, 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, str0 ∈ entries1, entries[str] ∈ cache_dirs, entries1[str0] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, 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)⟫ ⟨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)), entries[str] = ROOT_INO

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [cache_dirs[m3], m7, md6, m12, m13, entries1]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: cache_dirs[m3], m7, md6, m12, m13, entries1