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Proof step 10 from proof refine-rmdir (previous step)

cache_dirs[cache_dirs[m].entries[str]] = mkdir(m7, md1, m8, m9, entries), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(cache_dirs[m].entries[str], md0, true, m8, m9, m7), dent = mkdentry(str, cache_dirs[m].entries[str]), entries = ∅, err = err0, m0 = cache_dirs[m].nlink, m1 = cache_dirs[m].nsubdirs, m2 = cache_dirs[m].size, m3 = cache_dirs[m].entries[str], m4 = m8, m5 = m9, m6 = m7, 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), cache_dirs[m].entries[str] ∈ cache_dirs, str ∈ cache_dirs[m].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)⟫ ⟨dent0 := mkdentry(str, m3); p_inode0 := mkinode(m, md, true, m0, m1, m2); p_inode1 := mkinode(m3, md0, true, m4, m5, m6); 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), cache_dirs[m].entries[str] = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
# links(cache_dirs[m].entries[str], cache_dirs) = m8, cache_dirs[cache_dirs[m].entries[str]] = mkdir(m7, md1, # links(cache_dirs[m].entries[str], cache_dirs), m9, ∅), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = mkinode(cache_dirs[m].entries[str], md0, true, # links(cache_dirs[m].entries[str], cache_dirs), m9, m7), dent = mkdentry(str, cache_dirs[m].entries[str]), err = err0, m0 = cache_dirs[m].nlink, m1 = cache_dirs[m].nsubdirs, m2 = cache_dirs[m].size, m3 = cache_dirs[m].entries[str], p_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), cache_dirs[m].entries[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), cache_dirs[m].entries[str] ∈ cache_dirs, str ∈ cache_dirs[m].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)⟫ ⟨dent0 := mkdentry(str, m3); p_inode0 := mkinode(m, md, true, m0, m1, m2); p_inode1 := mkinode(m3, 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)

Rule argument

Additional information

Used simplifier rules: afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base