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Proof step 17 from proof refine-rmdir (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries[str], cache_dirs) = m1, cache_dirs[m].nlink = m5, cache_dirs[entries[str]] = mkdir(m0, md1, m1, m2, ∅), cache_dirs[m] = mkdir(cache_dirs[m].size, md2, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries), entries[str] = cache_dirs[m].entries[str], cache_dirs[entries[str]] = mkdir(m0, md3, m1, m2, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(entries[str], md0, true, m1, m2, m0), dent = mkdentry(str, entries[str]), entries0 = ∅, err = err0, m1 = m8, m4 = cache_dirs[m].size, m6 = cache_dirs[m].nsubdirs, m7 = m0, m9 = m2, p_inode = 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), m ∈ cache_dirs, str ∈ entries, entries[str] ∈ cache_dirs, str ∈ cache_dirs[m].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [cache_rmdir#(; p_inode, c_inode, 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) ∧ synced(cache_dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) ∧ afs-cons(cache_dirs, cache_files) → ⟨dent0 := mkdentry(str, entries[str]); p_inode0 := mkinode(m, md, true, m5, m6, m4); p_inode1 := mkinode(entries[str], md0, true, m8, m9, m7); afs_rmdir#(; p_inode0, p_inode1, dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0)), entries[str] = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
# links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries[str], cache_dirs) = m8, cache_dirs[m].nlink = m5, entries[str] = cache_dirs[m].entries[str], cache_dirs[m] = mkdir(cache_dirs[m].size, md2, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries), cache_dirs[entries[str]] = mkdir(m0, md3, # links(entries[str], cache_dirs), m2, ∅), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(entries[str], md0, true, # links(entries[str], cache_dirs), m2, m0), dent = mkdentry(str, entries[str]), err = err0, m4 = cache_dirs[m].size, m6 = cache_dirs[m].nsubdirs, p_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), entries[str] ≠ ROOT_INO, 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[m].entries, entries[str] ∈ cache_dirs, str ∈ entries, m ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [cache_rmdir#(; p_inode, c_inode, 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) ∧ synced(cache_dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) ∧ afs-cons(cache_dirs, cache_files) → ⟨dent0 := mkdentry(str, entries[str]); p_inode0 := mkinode(m, md, true, m5, m6, m4); p_inode1 := mkinode(entries[str], md0, true, m8, m2, m0); afs_rmdir#(; p_inode0, p_inode1, dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0))

Rule argument

Additional information

Used simplifier rules: ⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)