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Proof step 74 from proof refine-rename-file-dentry (previous step)

# links(entries0[str0], cache_dirs) = m7, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries[str], cache_dirs) = m5, cache_dirs[m].nlink = m16, cache_dirs[m3].nlink = m1, cache_files[entries[str]] = mkfile(m4, md7, # links(entries[str], cache_dirs), content1), entries0[str0] = cache_dirs[m3].entries[str0], entries[str] = cache_dirs[m].entries[str], cache_dirs[m3] = mkdir(cache_dirs[m3].size, md6, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m12, md3, # links(entries[str], cache_dirs), content), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str0]] = mkfile(m8, md4, # links(entries0[str0], cache_dirs), content0), cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries), cache_files[entries0[str0]] = mkfile(m6, md8, # links(entries0[str0], cache_dirs), content2), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m0 = cache_dirs[m3].size, m11 = cache_dirs[m].size, m17 = cache_dirs[m].nsubdirs, m2 = cache_dirs[m3].nsubdirs, new_child_inode = mkinode(entries[str], md1, false, # links(entries[str], cache_dirs), 0, m10), new_dent = mkdentry(str, entries[str]), new_parent_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), old_child_inode = mkinode(entries0[str0], md2, false, # links(entries0[str0], cache_dirs), 0, m13), old_dent = mkdentry(str0, entries0[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 ⊕ 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, str ∈ cache_dirs[m].entries, entries0[str0] ∈ cache_files, entries[str] ∈ cache_files, m3 ∈ cache_dirs, m ∈ cache_dirs, str0 ∈ entries0, str ∈ entries, 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, entries[str]); old_dent0 := mkdentry(str0, entries0[str0]); new_parent_inode0 := mkinode(m, md, true, m16, m17, m11); old_parent_inode0 := mkinode(m3, md0, true, m1, m2, m0); new_child_inode0 := mkinode(entries[str], md1, false, m5, 0, m10); old_child_inode0 := mkinode(entries0[str0], md2, false, m7, 0, m13); 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 75:
# links(entries[str], cache_dirs) = m5, # links(m, cache_dirs) = m14, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries0[str0], cache_dirs) = m7, cache_dirs[m3].nlink = m1, cache_files[entries0[str0]] = mkfile(m6, md8, # links(entries0[str0], cache_dirs), content2), cache_dirs[m] = mkdir(m9, md5, m14, m15, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str0]] = mkfile(m8, md4, # links(entries0[str0], cache_dirs), content0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m12, md3, # links(entries[str], cache_dirs), content), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md6, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries0), entries[str] = entries1[str], entries0[str0] = cache_dirs[m3].entries[str0], cache_files[entries[str]] = mkfile(m4, md7, # links(entries[str], cache_dirs), content1), cache_dirs[m] = mkdir(m9, md9, m14, m15, entries1), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m0 = cache_dirs[m3].size, m11 = m9, m14 = m16, m17 = m15, m2 = cache_dirs[m3].nsubdirs, new_child_inode = mkinode(entries[str], md1, false, # links(entries[str], cache_dirs), 0, m10), new_dent = mkdentry(str, entries[str]), new_parent_inode = mkinode(m, md, true, m14, m15, m9), old_child_inode = mkinode(entries0[str0], md2, false, # links(entries0[str0], cache_dirs), 0, m13), old_dent = mkdentry(str0, entries0[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 ∈ entries, str0 ∈ entries0, m ∈ cache_dirs, m3 ∈ cache_dirs, entries[str] ∈ cache_files, entries0[str0] ∈ cache_files, str ∈ entries1, 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, entries[str]); old_dent0 := mkdentry(str0, entries0[str0]); new_parent_inode0 := mkinode(m, md, true, m16, m17, m11); old_parent_inode0 := mkinode(m3, md0, true, m1, m2, m0); new_child_inode0 := mkinode(entries[str], md1, false, m5, 0, m10); old_child_inode0 := mkinode(entries0[str0], md2, false, m7, 0, m13); 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

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[m], m9, md9, m14, m15, 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[m], m9, md9, m14, m15, entries1