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Proof step 90 from proof refine-renam (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries2[str0], cache_dirs) = m7, # links(m3, cache_dirs) = m20, # links(entries0[str], cache_dirs) = m13, cache_dirs[m].nlink = m5, entries0[str] = cache_dirs[m].entries[str], cache_dirs[entries0[str]] = mkdir(m10, md5, # links(entries0[str], cache_dirs), m14, entries), cache_dirs[m3] = mkdir(m8, md8, # links(m3, cache_dirs), m11, entries2), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m15, md3, m16, content), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries2[str0]] = 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[entries2[str0]] = mkdir(m1, md9, # links(entries2[str0], cache_dirs), m9, ∅), 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(entries2[str0], md1, true, # links(entries2[str0], cache_dirs), m9, m1), new_dent = mkdentry(str0, entries2[str0]), new_parent_inode = mkinode(m3, md0, true, # links(m3, cache_dirs), m11, m8), old_child_inode = mkinode(entries0[str], md2, true, # links(entries0[str], cache_dirs), m14, m10), 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), entries2[str0] ≠ 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), str ∈ cache_dirs[m].entries, str0 ∈ entries2, m3 ∈ cache_dirs, str ∈ entries0, m ∈ cache_dirs, entries0[str] ∈ cache_dirs, entries2[str0] ∈ cache_dirs, m3 = m → md0 = md, 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)] ( ⟨new_dent1 := mkdentry(str0, entries2[str0]); old_dent2 := mkdentry(str, entries0[str]); new_child_inode1 := mkinode(entries2[str0], md1, true, m7, m9, m1); new_parent_inode1 := mkinode(m3, md0, true, m20, m11, m8); old_child_inode1 := mkinode(entries0[str], md2, true, m13, m14, m10); 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, m11, m8); p_inode2 := mkinode(entries2[str0], md1, true, m7, m9, m1); p_inode1 := mkinode(entries0[str], md2, true, m13, m14, m10); old_dent1 := mkdentry(str0, entries2[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 91:
# links(entries0[str], cache_dirs) = m13, # links(m3, cache_dirs) = m20, # links(entries2[str0], cache_dirs) = m7, # links(m, cache_dirs) = m2, cache_dirs[entries2[str0]] = mkdir(m1, md9, # links(entries2[str0], cache_dirs), m9, ∅), cache_dirs[m] = mkdir(m0, md6, m2, m12, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries2[str0]] = mkfile(m17, md4, m18, content0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m15, md3, m16, content), cache_dirs[m3] = mkdir(m8, md8, # links(m3, cache_dirs), m11, entries2), cache_dirs[entries0[str]] = mkdir(m10, md5, # links(entries0[str], cache_dirs), m14, entries), entries0[str] = entries1[str], cache_dirs[m] = mkdir(m0, md7, m2, m12, entries1), afs_dirs0 = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m2 = m5, m4 = m0, m6 = m12, new_child_inode = mkinode(entries2[str0], md1, true, # links(entries2[str0], cache_dirs), m9, m1), new_dent = mkdentry(str0, entries2[str0]), new_parent_inode = mkinode(m3, md0, true, # links(m3, cache_dirs), m11, m8), old_child_inode = mkinode(entries0[str], md2, true, # links(entries0[str], cache_dirs), m14, m10), old_dent = mkdentry(str, entries0[str]), old_parent_inode = mkinode(m, md, true, m2, m12, m0), old_parent_inode1 = mkinode(m, md, true, m2, m12, m0), 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), entries2[str0] ∈ cache_dirs, entries0[str] ∈ cache_dirs, m ∈ cache_dirs, str ∈ entries0, m3 ∈ cache_dirs, str0 ∈ entries2, str ∈ entries1, cache_sync → synced(cache_files, cache_icache, cache_pcache), m3 = m → md0 = md, 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_dent1 := mkdentry(str0, entries2[str0]); old_dent2 := mkdentry(str, entries0[str]); new_child_inode1 := mkinode(entries2[str0], md1, true, m7, m9, m1); new_parent_inode1 := mkinode(m3, md0, true, m20, m11, m8); old_child_inode1 := mkinode(entries0[str], md2, true, m13, m14, m10); 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, m11, m8); p_inode2 := mkinode(entries2[str0], md1, true, m7, m9, m1); p_inode1 := mkinode(entries0[str], md2, true, m13, m14, m10); old_dent1 := mkdentry(str0, entries2[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)), entries2[str0] = 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[m], m0, md7, m2, m12, 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], m0, md7, m2, m12, entries1