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Proof step 5 from proof refine-rmdir (previous step)

cache_dirs[p_inode1.ino].entries = ∅, p_inode1.ino = cache_dirs[p_inode0.ino].entries[str], p_inode0.nlink = cache_dirs[p_inode0.ino].nlink, p_inode1.nlink = cache_dirs[p_inode1.ino].nlink, p_inode0.nsubdirs = cache_dirs[p_inode0.ino].nsubdirs, p_inode1.nsubdirs = cache_dirs[p_inode1.ino].nsubdirs, p_inode0.size = cache_dirs[p_inode0.ino].size, p_inode1.size = cache_dirs[p_inode1.ino].size, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = p_inode1, dent = mkdentry(str, p_inode1.ino), err = err0, p_inode = p_inode0, p_inode1.ino ≠ ROOT_INO, p_inode0.directory, p_inode1.directory, 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), p_inode0.ino ∈ cache_dirs, str ∈ cache_dirs[p_inode0.ino].entries, p_inode1.ino ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(p_inode0.ino, cache_dirs) ≠ ∅ ∨ p_inode0.ino = ROOT_INO ⊦ ⟪cache_rmdir#(; p_inode, c_inode, dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨dent0 := mkdentry(str, p_inode1.ino); 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)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 6:
cache_dirs[p_inode1.ino].entries = ∅, p_inode1.ino = cache_dirs[m].entries[str], p_inode1.nlink = cache_dirs[p_inode1.ino].nlink, p_inode1.nsubdirs = cache_dirs[p_inode1.ino].nsubdirs, p_inode1.size = cache_dirs[p_inode1.ino].size, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, boolvar, c_inode = p_inode1, dent = mkdentry(str, p_inode1.ino), err = err0, m0 = cache_dirs[m].nlink, m1 = cache_dirs[m].nsubdirs, m2 = cache_dirs[m].size, p_inode = p_inode0, p_inode0 = mkinode(m, md, boolvar, m0, m1, m2), p_inode1.directory, 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), p_inode1.ino ∈ 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, p_inode1.ino); 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), p_inode1.ino = ROOT_INO

Rule argument

spec: inode name: elim seq: ⊦ ( ( ( (m = inode.ino ∧ md = inode.meta) ∧ (boolvar ↔ inode.directory)) ∧ m0 = inode.nlink) ∧ m1 = inode.nsubdirs) ∧ m2 = inode.size ↔ inode = mkinode(m, md, boolvar, m0, m1, m2) subst: [inode, m, md, boolvar, m0, m1, m2] → [p_inode0, m, md, boolvar, m0, m1, m2]

Additional information

Used simplifier rules: ⊦ ( ( ( (m = inode.ino ∧ md = inode.meta) ∧ (boolvar ↔ inode.directory)) ∧ m0 = inode.nlink) ∧ m1 = inode.nsubdirs) ∧ m2 = inode.size ↔ inode = mkinode(m, md, boolvar, m0, m1, m2); (elim) Used terms: p_inode0, m, md, boolvar, m0, m1, m2