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Proof step 33 from proof refine-rename-file-negdentry (previous step)

# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries[str], cache_dirs) = cache_files[entries[str]].nlink, cache_dirs[m3].nlink = m1, cache_dirs[m].nlink = m12, cache_files[entries[str]].nlink = m5, entries[str] = cache_dirs[m3].entries[str], cache_files[entries[str]] = mkfile(m4, md5, cache_files[entries[str]].nlink, content0), cache_dirs[m] = mkdir(cache_dirs[m].size, md6, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m9, md2, cache_files[entries[str]].nlink, content), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md3, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m11 = cache_dirs[m].nsubdirs, m13 = cache_dirs[m3].nsubdirs, m6 = cache_dirs[m].size, m7 = cache_dirs[m3].size, new_child_inode = new_child_inode0, new_dent = negdentry(str0), new_parent_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), old_child_inode = mkinode(entries[str], md1, false, cache_files[entries[str]].nlink, 0, m10), old_dent = mkdentry(str, entries[str]), 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), str ∈ cache_dirs[m3].entries, str ∈ entries, m ∈ cache_dirs, entries[str] ∈ cache_files, m3 ∈ cache_dirs, ¬ str0 ∈ cache_dirs[m].entries, ¬ str0 ∈ entries1, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), 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)] ( afs-cons(cache_dirs, cache_files) → ⟪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); skip; 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)}⟫ ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m, md, true, m12, m11, m6); old_parent_inode0 := mkinode(m3, md0, true, m1, m13, m7); old_child_inode0 := mkinode(entries[str], md1, false, m5, 0, m10); 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 34:
# links(entries[str], cache_dirs) = m2, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, cache_dirs[m].nlink = m12, cache_dirs[m3].nlink = m1, cache_dirs[m3] = mkdir(cache_dirs[m3].size, md3, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m9, md2, m2, content), cache_dirs[m] = mkdir(cache_dirs[m].size, md6, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), cache_files[entries[str]] = mkfile(m4, md5, m2, content0), entries[str] = cache_dirs[m3].entries[str], cache_files[entries[str]] = mkfile(m0, md4, m2, content1), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m11 = cache_dirs[m].nsubdirs, m13 = cache_dirs[m3].nsubdirs, m2 = m5, m6 = cache_dirs[m].size, m7 = cache_dirs[m3].size, new_child_inode = new_child_inode0, new_dent = negdentry(str0), new_parent_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), old_child_inode = mkinode(entries[str], md1, false, m2, 0, m10), old_dent = mkdentry(str, entries[str]), 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), m3 ∈ cache_dirs, entries[str] ∈ cache_files, m ∈ cache_dirs, str ∈ entries, str ∈ cache_dirs[m3].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, 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)] ( afs-cons(cache_dirs, cache_files) → ⟪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); skip; 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)}⟫ ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m, md, true, m12, m11, m6); old_parent_inode0 := mkinode(m3, md0, true, m1, m13, m7); old_child_inode0 := mkinode(entries[str], md1, false, m5, 0, m10); 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))), str0 ∈ cache_dirs[m].entries, str0 ∈ entries1

Rule argument

spec: abstract-file name: elim seq: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content) subst: [af, m, md, m0, content] → [cache_files[entries[str]], m0, md4, m2, content1]

Additional information

Used simplifier rules: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content); (elim) Used terms: cache_files[entries[str]], m0, md4, m2, content1