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

afs_asynced ↔ cache_asynced, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = p_inode1, err = err0, new_dent = old_dent0, p_inode = p_inode0, 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-link(p_inode, c_inode, old_dent, new_dent, afs_dirs, afs_files), synced(cache_dirs, cache_icache, cache_pcache), cache_sync → synced(cache_files, cache_icache, cache_pcache) ⊦ ⟪cache_link#(old_dent; p_inode, c_inode, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨afs_link#(old_dent; p_inode0, p_inode1, old_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 ∧ new_dent = old_dent0 ∧ err = err0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
p_inode1.ino = old_dent.ino, old_dent.ino = cache_dirs[ino].entries[old_dent.name], p_inode1.nlink = ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[old_dent.ino].nlink, p_inode0.nlink = cache_dirs[p_inode0.ino].nlink, p_inode1.nsubdirs = 0, p_inode0.nsubdirs = cache_dirs[p_inode0.ino].nsubdirs, p_inode0.size = cache_dirs[p_inode0.ino].size, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = p_inode1, err = err0, new_dent = old_dent0, p_inode = p_inode0, old_dent.dentry?, p_inode0.directory, old_dent0.negdentry?, 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), old_dent.ino ∈ cache_files, p_inode0.ino ∈ cache_dirs, old_dent.name ∈ cache_dirs[ino].entries, ino ∈ cache_dirs, ¬ p_inode1.directory, ¬ old_dent0.name ∈ cache_dirs[p_inode0.ino].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(old_dent.ino, cache_dirs) ≠ ∅ ∨ old_dent.ino = ROOT_INO, links(p_inode0.ino, cache_dirs) ≠ ∅ ∨ p_inode0.ino = ROOT_INO, links(ino, cache_dirs) ≠ ∅ ∨ ino = ROOT_INO ⊦ ⟪cache_link#(old_dent; p_inode, c_inode, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨afs_link#(old_dent; p_inode0, p_inode1, old_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 ∧ new_dent = old_dent0 ∧ err = err0)

Rule argument

Additional information

Used simplifier rules: ⊦ 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-negdentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ ¬ dent.name ∈ dirs[ino].entries ∧ dent.negdentry?; (s)% ⊦ ino ∈ files ⊕ pcache ↔ ino ∈ files; (s)% ⊦ ino ∈ files ⊕ icache ↔ ino ∈ files; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (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-dentry(dent, dirs, files) ↔ (∃ ino. valid-dentry(ino, dent, dirs, files)); (s)%⊦ pre-link(p_inode, c_inode, old_dent, new_dent, dirs, files) ↔ valid-dentry(old_dent, dirs, files) ∧ valid-negdentry(p_inode.ino, new_dent, dirs, files) ∧ (links(old_dent.ino, dirs) ≠ ∅ ∨ old_dent.ino = ROOT_INO) ∧ valid-dir-inode(p_inode, dirs, files) ∧ valid-file-inode(c_inode, dirs, files) ∧ c_inode.ino = old_dent.ino; (s)