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Proof step 58 from proof refine-renam (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = m20, cache_dirs[m].nlink = m5, entries0[str] = cache_dirs[m].entries[str], cache_dirs[entries0[str]] = mkdir(m0, md7, m1, m2, entries1), cache_dirs[m3] = mkdir(m12, md9, # links(m3, cache_dirs), m24, entries3), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m15, md3, m16, content), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[negdentry(str0).ino] = mkfile(m17, md4, m18, content0), cache_dirs[m] = mkdir(cache_dirs[m].size, md6, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_dirs[negdentry(str0).ino] = mkdir(m11, md8, m22, m23, entries2), afs_dirs0 = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m4 = cache_dirs[m].size, m6 = cache_dirs[m].nsubdirs, new_child_inode = mkinode(m7, md1, boolvar, m8, m9, m10), new_dent = negdentry(str0), new_parent_inode = mkinode(m3, md0, true, # links(m3, cache_dirs), m24, m12), old_child_inode = mkinode(entries0[str], md2, false, m16, 0, m14), old_dent = mkdentry(str, entries0[str]), old_parent_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), old_parent_inode1 = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].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 ∈ cache_dirs[m].entries, m3 ∈ cache_dirs, str ∈ entries0, m ∈ cache_dirs, entries0[str] ∈ cache_files, ¬ entries0[str] ∈ cache_dirs, ¬ str0 ∈ entries3, m3 = m → md0 = md, 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_child_inode1 := mkinode(m7, md1, boolvar, m8, m9, m10); new_dent1 := negdentry(str0); old_child_inode1 := mkinode(entries0[str], md2, false, m16, 0, m14); old_dent2 := mkdentry(str, entries0[str]); new_parent_inode1 := mkinode(m3, md0, true, m20, m24, m12); afs_rename# (; old_parent_inode1, new_parent_inode1, old_child_inode1, new_child_inode1, old_dent2, new_dent1, err0, afs_dirs0, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode1 ∧ new_parent_inode = new_parent_inode1 ∧ old_child_inode = old_child_inode1 ∧ new_child_inode = new_child_inode1 ∧ old_dent = old_dent2 ∧ new_dent = new_dent1 ∧ err = err0 ∧ afs_dirs0 = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced)) → ⟨old_dent0 := mkdentry(str, entries0[str]); p_inode := mkinode(m, md, true, m5, m6, m4); p_inode0 := mkinode(m3, md0, true, m20, m24, m12); p_inode2 := mkinode(m7, md1, boolvar, m8, m9, m10); p_inode1 := mkinode(entries0[str], md2, false, m16, 0, m14); old_dent1 := negdentry(str0); afs_rename#(; p_inode, p_inode0, p_inode1, p_inode2, old_dent0, old_dent1, err0, afs_dirs0, afs_files; afs_asynced)⟩ ( afs_dirs0 = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ old_parent_inode = p_inode ∧ new_parent_inode = p_inode0 ∧ old_child_inode = p_inode1 ∧ new_child_inode = p_inode2 ∧ old_dent = old_dent0 ∧ new_dent = old_dent1 ∧ err = err0))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 59:
# links(m3, cache_dirs) = m20, # links(m, cache_dirs) = m19, cache_dirs[negdentry(str0).ino] = mkdir(m11, md8, m22, m23, entries2), cache_dirs[m] = mkdir(m13, md6, m19, m21, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[negdentry(str0).ino] = mkfile(m17, md4, m18, content0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m15, md3, m16, content), cache_dirs[m3] = mkdir(m12, md9, # links(m3, cache_dirs), m24, entries3), cache_dirs[entries0[str]] = mkdir(m0, md7, m1, m2, entries1), entries0[str] = entries[str], cache_dirs[m] = mkdir(m13, md5, m19, m21, entries), afs_dirs0 = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m19 = m5, m4 = m13, m6 = m21, new_child_inode = mkinode(m7, md1, boolvar, m8, m9, m10), new_dent = negdentry(str0), new_parent_inode = mkinode(m3, md0, true, # links(m3, cache_dirs), m24, m12), old_child_inode = mkinode(entries0[str], md2, false, m16, 0, m14), old_dent = mkdentry(str, entries0[str]), old_parent_inode = mkinode(m, md, true, m19, m21, m13), old_parent_inode1 = mkinode(m, md, true, m19, m21, m13), 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), entries0[str] ∈ cache_files, m ∈ cache_dirs, str ∈ entries0, m3 ∈ cache_dirs, str ∈ entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), m3 = m → md0 = md, 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_child_inode1 := mkinode(m7, md1, boolvar, m8, m9, m10); new_dent1 := negdentry(str0); old_child_inode1 := mkinode(entries0[str], md2, false, m16, 0, m14); old_dent2 := mkdentry(str, entries0[str]); new_parent_inode1 := mkinode(m3, md0, true, m20, m24, m12); afs_rename# (; old_parent_inode1, new_parent_inode1, old_child_inode1, new_child_inode1, old_dent2, new_dent1, err0, afs_dirs0, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode1 ∧ new_parent_inode = new_parent_inode1 ∧ old_child_inode = old_child_inode1 ∧ new_child_inode = new_child_inode1 ∧ old_dent = old_dent2 ∧ new_dent = new_dent1 ∧ err = err0 ∧ afs_dirs0 = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced)) → ⟨old_dent0 := mkdentry(str, entries0[str]); p_inode := mkinode(m, md, true, m5, m6, m4); p_inode0 := mkinode(m3, md0, true, m20, m24, m12); p_inode2 := mkinode(m7, md1, boolvar, m8, m9, m10); p_inode1 := mkinode(entries0[str], md2, false, m16, 0, m14); old_dent1 := negdentry(str0); afs_rename#(; p_inode, p_inode0, p_inode1, p_inode2, old_dent0, old_dent1, err0, afs_dirs0, afs_files; afs_asynced)⟩ ( afs_dirs0 = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ old_parent_inode = p_inode ∧ new_parent_inode = p_inode0 ∧ old_child_inode = p_inode1 ∧ new_child_inode = p_inode2 ∧ old_dent = old_dent0 ∧ new_dent = old_dent1 ∧ err = err0)), entries0[str] ∈ cache_dirs, str0 ∈ entries3

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], m13, md5, m19, m21, 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[m], m13, md5, m19, m21, entries