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Proof step 38 from proof refine-rename-dir-dentry (previous step)

# links(entries0[str0], cache_dirs) = cache_dirs[entries0[str0]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries2[str], cache_dirs) = cache_dirs[entries2[str]].nlink, cache_dirs[entries2[str]].entries = ∅, cache_dirs[m3].nlink = m12, cache_dirs[entries0[str0]].nlink = m1, cache_dirs[m].nlink = m9, cache_dirs[entries2[str]].nlink = m16, entries0[str0] = cache_dirs[m3].entries[str0], entries2[str] = cache_dirs[m].entries[str], cache_dirs[entries0[str0]] = mkdir(cache_dirs[entries0[str0]].size, md4, cache_dirs[entries0[str0]].nlink, cache_dirs[entries0[str0]].nsubdirs, entries), cache_dirs[entries2[str]] = mkdir(cache_dirs[entries2[str]].size, md7, cache_dirs[entries2[str]].nlink, cache_dirs[entries2[str]].nsubdirs, ∅), cache_dirs[m] = mkdir(cache_dirs[m].size, md3, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries2), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md5, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m0 = cache_dirs[m3].size, m10 = cache_dirs[m3].nsubdirs, m11 = cache_dirs[entries0[str0]].nsubdirs, m14 = cache_dirs[m].nsubdirs, m4 = cache_dirs[entries2[str]].size, m5 = cache_dirs[m].size, m6 = cache_dirs[entries2[str]].nsubdirs, m8 = cache_dirs[entries0[str0]].size, new_child_inode = mkinode(entries2[str], md1, true, cache_dirs[entries2[str]].nlink, cache_dirs[entries2[str]].nsubdirs, cache_dirs[entries2[str]].size), new_dent = mkdentry(str, entries2[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, true, cache_dirs[entries0[str0]].nlink, cache_dirs[entries0[str0]].nsubdirs, cache_dirs[entries0[str0]].size), 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), entries2[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), str0 ∈ cache_dirs[m3].entries, str ∈ cache_dirs[m].entries, str ∈ entries2, str0 ∈ entries0, entries2[str] ∈ cache_dirs, entries0[str0] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, 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 ⊦ [afs_rename#(; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files; asynced1); if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, cache_icache); icache_set#(new_parent_inode, asynced; err, cache_icache); if mkdentry(str, entries2[str]).dentry? then icache_set#(new_child_inode, asynced; err, cache_icache); dcache_delete#(old_parent_inode.ino, old_dent.name; err, cache_dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, cache_dcache)}] ( 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, entries0[str0]); new_parent_inode0 := mkinode(m, md, true, m9, m14, m5); old_parent_inode0 := mkinode(m3, md0, true, m12, m10, m0); new_child_inode0 := mkinode(entries2[str], md1, true, m16, m6, m4); old_child_inode0 := mkinode(entries0[str0], md2, true, m1, m11, m8); 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 ↔ asynced1)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 39:
# links(entries2[str], cache_dirs) = m7, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries0[str0], cache_dirs) = cache_dirs[entries0[str0]].nlink, cache_dirs[m].nlink = m9, cache_dirs[entries0[str0]].nlink = m1, cache_dirs[m3].nlink = m12, cache_dirs[m3] = mkdir(cache_dirs[m3].size, md5, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries0), cache_dirs[m] = mkdir(cache_dirs[m].size, md3, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries2), cache_dirs[entries2[str]] = mkdir(m2, md7, m7, m13, ∅), cache_dirs[entries0[str0]] = mkdir(cache_dirs[entries0[str0]].size, md4, cache_dirs[entries0[str0]].nlink, cache_dirs[entries0[str0]].nsubdirs, entries), entries2[str] = cache_dirs[m].entries[str], entries0[str0] = cache_dirs[m3].entries[str0], cache_dirs[entries2[str]] = mkdir(m2, md6, m7, m13, entries1), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, entries1 = ∅, err = err0, m0 = cache_dirs[m3].size, m10 = cache_dirs[m3].nsubdirs, m11 = cache_dirs[entries0[str0]].nsubdirs, m14 = cache_dirs[m].nsubdirs, m4 = m2, m5 = cache_dirs[m].size, m6 = m13, m7 = m16, m8 = cache_dirs[entries0[str0]].size, new_child_inode = mkinode(entries2[str], md1, true, m7, m13, m2), new_dent = mkdentry(str, entries2[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, true, cache_dirs[entries0[str0]].nlink, cache_dirs[entries0[str0]].nsubdirs, cache_dirs[entries0[str0]].size), 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), m3 ∈ cache_dirs, m ∈ cache_dirs, entries0[str0] ∈ cache_dirs, entries2[str] ∈ cache_dirs, str0 ∈ entries0, str ∈ entries2, str ∈ cache_dirs[m].entries, str0 ∈ cache_dirs[m3].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 ⊦ [afs_rename#(; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files; asynced1); if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, cache_icache); icache_set#(new_parent_inode, asynced; err, cache_icache); if mkdentry(str, entries2[str]).dentry? then icache_set#(new_child_inode, asynced; err, cache_icache); dcache_delete#(old_parent_inode.ino, old_dent.name; err, cache_dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, cache_dcache)}] ( 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, entries0[str0]); new_parent_inode0 := mkinode(m, md, true, m9, m14, m5); old_parent_inode0 := mkinode(m3, md0, true, m12, m10, m0); new_child_inode0 := mkinode(entries2[str], md1, true, m16, m6, m4); old_child_inode0 := mkinode(entries0[str0], md2, true, m1, m11, m8); 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 ↔ asynced1))), 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[entries2[str]], m2, md6, m7, 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[entries2[str]], m2, md6, m7, m13, entries1