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Proof step 2 from proof refine-renam (previous step)

afs_asynced ↔ cache_asynced, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, new_child_inode = p_inode2, new_dent = old_dent1, new_parent_inode = p_inode0, old_child_inode = p_inode1, old_dent = old_dent0, old_parent_inode = p_inode, afs-cons(cache_dirs, cache_files), afs-cons(afs_dirs, afs_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), pre-rename(old_parent_inode, old_dent, old_child_inode, new_parent_inode, new_dent, new_child_inode, afs_dirs, afs_files), synced(cache_dirs, cache_icache, cache_pcache), cache_sync → synced(cache_files, cache_icache, cache_pcache) ⊦ ⟪cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨afs_rename#(; p_inode, p_inode0, p_inode1, p_inode2, old_dent0, old_dent1, err0, afs_dirs, afs_files; afs_asynced)⟩ ( (afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced)) ∧ old_parent_inode = p_inode ∧ new_parent_inode = p_inode0 ∧ old_child_inode = p_inode1 ∧ new_child_inode = p_inode2 ∧ old_dent = old_dent0 ∧ new_dent = old_dent1 ∧ err = err0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
old_dent0.ino = cache_dirs[p_inode.ino].entries[old_dent0.name], p_inode1.ino = old_dent0.ino, p_inode.nlink = cache_dirs[p_inode.ino].nlink, p_inode0.nlink = cache_dirs[p_inode0.ino].nlink, p_inode.nsubdirs = cache_dirs[p_inode.ino].nsubdirs, p_inode0.nsubdirs = cache_dirs[p_inode0.ino].nsubdirs, p_inode.size = cache_dirs[p_inode.ino].size, p_inode0.size = cache_dirs[p_inode0.ino].size, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, new_child_inode = p_inode2, new_dent = old_dent1, new_parent_inode = p_inode0, old_child_inode = p_inode1, old_dent = old_dent0, old_parent_inode = p_inode, old_dent0.dentry?, p_inode.directory, p_inode0.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_inode.ino ∈ cache_dirs, old_dent0.name ∈ cache_dirs[p_inode.ino].entries, p_inode0.ino ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), p_inode0.ino = p_inode.ino → p_inode0.meta = p_inode.meta, (links(p_inode0.ino, cache_dirs) ≠ ∅ ∨ p_inode0.ino = ROOT_INO) ∧ ¬ old_dent1.name ∈ cache_dirs[p_inode0.ino].entries ∧ old_dent1.negdentry? ∨ (links(p_inode0.ino, cache_dirs) ≠ ∅ ∨ p_inode0.ino = ROOT_INO) ∧ (old_dent1.ino ∈ cache_dirs ∨ old_dent1.ino ∈ cache_files) ∧ old_dent1.name ∈ cache_dirs[p_inode0.ino].entries ∧ old_dent1.ino = cache_dirs[p_inode0.ino].entries[old_dent1.name] ∧ old_dent1.dentry? ∧ ( p_inode2.ino ∈ cache_dirs ∧ p_inode2.nlink = cache_dirs[p_inode2.ino].nlink ∧ p_inode2.size = cache_dirs[p_inode2.ino].size ∧ p_inode2.nsubdirs = cache_dirs[p_inode2.ino].nsubdirs ∧ p_inode2.directory ∨ p_inode2.ino ∈ cache_files ∧ p_inode2.nlink = ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[p_inode2.ino].nlink ∧ p_inode2.nsubdirs = 0 ∧ ¬ p_inode2.directory) ∧ p_inode2.ino = old_dent1.ino ∧ (old_dent1.ino ∈ cache_files ↔ old_dent0.ino ∈ cache_files) ∧ (old_dent1.ino ∈ cache_dirs → old_dent1.ino ≠ ROOT_INO ∧ cache_dirs[old_dent1.ino].entries = ∅), links(p_inode.ino, cache_dirs) ≠ ∅ ∨ p_inode.ino = ROOT_INO, old_dent0.ino ∈ cache_dirs ∨ old_dent0.ino ∈ cache_files, old_dent0.ino ∈ cache_dirs ∧ p_inode1.nlink = cache_dirs[old_dent0.ino].nlink ∧ p_inode1.size = cache_dirs[old_dent0.ino].size ∧ p_inode1.nsubdirs = cache_dirs[old_dent0.ino].nsubdirs ∧ p_inode1.directory ∨ old_dent0.ino ∈ cache_files ∧ p_inode1.nlink = ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[old_dent0.ino].nlink ∧ p_inode1.nsubdirs = 0 ∧ ¬ p_inode1.directory ⊦ ⟪cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨afs_rename#(; p_inode, p_inode0, p_inode1, p_inode2, old_dent0, old_dent1, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ old_parent_inode = p_inode ∧ new_parent_inode = p_inode0 ∧ old_child_inode = p_inode1 ∧ new_child_inode = p_inode2 ∧ old_dent = old_dent0 ∧ new_dent = old_dent1 ∧ err = err0)

Rule argument

Additional information

Used simplifier rules: ⊦ dir-empty(ino, dirs, files) ↔ dirs[ino].entries = ∅; (s)% ⊦ valid-file-ino(ino, dirs, files) ↔ ino ∈ files; (s)%⊦ valid-file-inode(inode, dirs, files) ↔ valid-file-ino(inode.ino, dirs, files) ∧ inode.nlink = files[inode.ino].nlink ∧ inode.nsubdirs = 0 ∧ ¬ inode.directory; (s)% ⊦ valid-dir-ino(ino, dirs, files) ↔ ino ∈ dirs; (s)%⊦ valid-dir-inode(inode, dirs, files) ↔ valid-dir-ino(inode.ino, dirs, files) ∧ inode.nlink = dirs[inode.ino].nlink ∧ inode.size = dirs[inode.ino].size ∧ inode.nsubdirs = dirs[inode.ino].nsubdirs ∧ inode.directory; (s)%⊦ valid-wb-inode(inode, dirs, files) ↔ valid-dir-inode(inode, dirs, files) ∨ valid-file-inode(inode, dirs, files); (s)% ⊦ ino ∈ files ⊕ pcache ↔ ino ∈ files; (s)% ⊦ ino ∈ files ⊕ icache ↔ ino ∈ files; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)%⊦ valid-dentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ valid-ino(dent.ino, dirs, files) ∧ dent.name ∈ dirs[ino].entries ∧ dent.ino = dirs[ino].entries[dent.name] ∧ dent.dentry?; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (s)%⊦ valid-negdentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ ¬ dent.name ∈ dirs[ino].entries ∧ dent.negdentry?; (s)%⊦ pre-rename (old_parent_inode, old_dent, old_child_inode, new_parent_inode, new_dent, new_child_inode, dirs, files) ↔ ( valid-negdentry(new_parent_inode.ino, new_dent, dirs, files) ∨ valid-dentry(new_parent_inode.ino, new_dent, dirs, files) ∧ valid-wb-inode(new_child_inode, dirs, files) ∧ new_child_inode.ino = new_dent.ino ∧ ( valid-file-ino(new_dent.ino, dirs, files) ↔ valid-file-ino(old_dent.ino, dirs, files)) ∧ ( valid-dir-ino(new_dent.ino, dirs, files) → new_dent.ino ≠ ROOT_INO ∧ dir-empty(new_dent.ino, dirs, files))) ∧ valid-dentry(old_parent_inode.ino, old_dent, dirs, files) ∧ valid-dir-inode(old_parent_inode, dirs, files) ∧ valid-dir-inode(new_parent_inode, dirs, files) ∧ ( new_parent_inode.ino = old_parent_inode.ino → new_parent_inode.meta = old_parent_inode.meta) ∧ valid-wb-inode(old_child_inode, dirs, files) ∧ old_child_inode.ino = old_dent.ino; (s)