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Proof step 30 from proof refine-rename-dir-dentry (previous step)

# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries1[str0], cache_dirs) = cache_dirs[entries1[str0]].nlink, # links(entries2[str], cache_dirs) = m16, # links(m, cache_dirs) = m9, cache_dirs[entries1[str0]].nlink = m1, cache_dirs[m3].nlink = m12, entries1[str0] = cache_dirs[m3].entries[str0], cache_dirs[m3] = mkdir(cache_dirs[m3].size, md6, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), cache_dirs[m] = mkdir(m5, md3, # links(m, cache_dirs), m14, entries2), cache_dirs[entries2[str]] = mkdir(m4, md7, # links(entries2[str], cache_dirs), m6, ∅), cache_dirs[entries1[str0]] = mkdir(cache_dirs[entries1[str0]].size, md5, cache_dirs[entries1[str0]].nlink, cache_dirs[entries1[str0]].nsubdirs, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m0 = cache_dirs[entries1[str0]].size, m13 = cache_dirs[m3].nsubdirs, m2 = cache_dirs[entries1[str0]].nsubdirs, m7 = cache_dirs[m3].size, new_child_inode = mkinode(entries2[str], md1, true, # links(entries2[str], cache_dirs), m6, m4), new_dent = mkdentry(str, entries2[str]), new_parent_inode = mkinode(m, md, true, # links(m, cache_dirs), m14, m5), old_child_inode = mkinode(entries1[str0], md2, true, cache_dirs[entries1[str0]].nlink, cache_dirs[entries1[str0]].nsubdirs, cache_dirs[entries1[str0]].size), old_dent = mkdentry(str0, entries1[str0]), old_parent_inode = mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size), entries2[str] ≠ ROOT_INO, 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), str0 ∈ cache_dirs[m3].entries, m3 ∈ cache_dirs, m ∈ cache_dirs, entries1[str0] ∈ cache_dirs, entries2[str] ∈ cache_dirs, str0 ∈ entries1, str ∈ entries2, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), 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)) → ⟨new_dent0 := mkdentry(str, entries2[str]); old_dent0 := mkdentry(str0, entries1[str0]); new_parent_inode0 := mkinode(m, md, true, m9, m14, m5); old_parent_inode0 := mkinode(m3, md0, true, m12, m13, m7); new_child_inode0 := mkinode(entries2[str], md1, true, m16, m6, m4); old_child_inode0 := mkinode(entries1[str0], 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 31:
# links(m, cache_dirs) = m9, # links(entries2[str], cache_dirs) = m16, # links(entries1[str0], cache_dirs) = m10, # links(m3, cache_dirs) = cache_dirs[m3].nlink, cache_dirs[m3].nlink = m12, cache_dirs[entries1[str0]] = mkdir(m8, md5, m10, m11, entries0), cache_dirs[entries2[str]] = mkdir(m4, md7, # links(entries2[str], cache_dirs), m6, ∅), cache_dirs[m] = mkdir(m5, md3, # links(m, cache_dirs), m14, entries2), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md6, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), entries1[str0] = cache_dirs[m3].entries[str0], cache_dirs[entries1[str0]] = mkdir(m8, md4, m10, m11, entries), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m0 = m8, m10 = m1, m13 = cache_dirs[m3].nsubdirs, m2 = m11, m7 = cache_dirs[m3].size, new_child_inode = mkinode(entries2[str], md1, true, # links(entries2[str], cache_dirs), m6, m4), new_dent = mkdentry(str, entries2[str]), new_parent_inode = mkinode(m, md, true, # links(m, cache_dirs), m14, m5), old_child_inode = mkinode(entries1[str0], md2, true, m10, m11, m8), old_dent = mkdentry(str0, entries1[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), str ∈ entries2, str0 ∈ entries1, entries2[str] ∈ cache_dirs, entries1[str0] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, str0 ∈ cache_dirs[m3].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, 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)) → ⟨new_dent0 := mkdentry(str, entries2[str]); old_dent0 := mkdentry(str0, entries1[str0]); new_parent_inode0 := mkinode(m, md, true, m9, m14, m5); old_parent_inode0 := mkinode(m3, md0, true, m12, m13, m7); new_child_inode0 := mkinode(entries2[str], md1, true, m16, m6, m4); old_child_inode0 := mkinode(entries1[str0], 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))), entries2[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[entries1[str0]], m8, md4, m10, m11, entries]

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[entries1[str0]], m8, md4, m10, m11, entries