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Proof step 5 from proof inv-rename-negdentry (previous step)

old_child_inode.ino = dirs[old_parent_inode.ino].entries[str], old_parent_inode.nlink = dirs[old_parent_inode.ino].nlink, new_parent_inode.nlink = dirs[new_parent_inode.ino].nlink, old_parent_inode.nsubdirs = dirs[old_parent_inode.ino].nsubdirs, new_parent_inode.nsubdirs = dirs[new_parent_inode.ino].nsubdirs, old_parent_inode.size = dirs[old_parent_inode.ino].size, new_parent_inode.size = dirs[new_parent_inode.ino].size, old_dent = mkdentry(str, old_child_inode.ino), old_parent_inode.directory, new_parent_inode.directory, new_dent.negdentry?, afs-cons(dirs, files), dcache-cons(dcache, dirs, files), icache-cons(icache, dirs, files), pcache-cons(pcache, dirs, files), synced(dirs, icache, pcache), new_parent_inode.ino ∈ dirs, old_parent_inode.ino ∈ dirs, str ∈ dirs[old_parent_inode.ino].entries, ¬ new_dent.name ∈ dirs[new_parent_inode.ino].entries, asynced → synced(files, icache, pcache), new_parent_inode.ino = old_parent_inode.ino → new_parent_inode.meta = old_parent_inode.meta, old_child_inode.ino ∈ dirs ∧ old_child_inode.nlink = dirs[old_child_inode.ino].nlink ∧ old_child_inode.size = dirs[old_child_inode.ino].size ∧ old_child_inode.nsubdirs = dirs[old_child_inode.ino].nsubdirs ∧ old_child_inode.directory ∨ old_child_inode.ino ∈ files ∧ old_child_inode.nlink = files[old_child_inode.ino].nlink ∧ old_child_inode.nsubdirs = 0 ∧ ¬ old_child_inode.directory, links(new_parent_inode.ino, dirs) ≠ ∅ ∨ new_parent_inode.ino = ROOT_INO, links(old_parent_inode.ino, dirs) ≠ ∅ ∨ old_parent_inode.ino = ROOT_INO, old_child_inode.ino ∈ dirs ∨ old_child_inode.ino ∈ files ⊦ ⟪cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, dirs, files, icache, dcache, asynced0)⟫ ( icache-cons(icache, dirs, files) ∧ dcache-cons(dcache, dirs, files) ∧ pcache-cons(pcache, dirs, files) ∧ afs-cons(dirs, files) ∧ synced(dirs, icache, pcache) ∧ (asynced → synced(files, icache, pcache)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 6:
old_child_inode.ino = dirs[old_parent_inode.ino].entries[str], new_parent_inode.nlink = dirs[new_parent_inode.ino].nlink, old_parent_inode.nlink = dirs[old_parent_inode.ino].nlink, new_parent_inode.nsubdirs = dirs[new_parent_inode.ino].nsubdirs, old_parent_inode.nsubdirs = dirs[old_parent_inode.ino].nsubdirs, new_parent_inode.size = dirs[new_parent_inode.ino].size, old_parent_inode.size = dirs[old_parent_inode.ino].size, new_dent = negdentry(str0), old_dent = mkdentry(str, old_child_inode.ino), new_parent_inode.directory, old_parent_inode.directory, new_dent.negdentry?, afs-cons(dirs, files), dcache-cons(dcache, dirs, files), icache-cons(icache, dirs, files), pcache-cons(pcache, dirs, files), synced(dirs, icache, pcache), str ∈ dirs[old_parent_inode.ino].entries, old_parent_inode.ino ∈ dirs, new_parent_inode.ino ∈ dirs, new_parent_inode.ino = old_parent_inode.ino → new_parent_inode.meta = old_parent_inode.meta, asynced → synced(files, icache, pcache), old_child_inode.ino ∈ dirs ∨ old_child_inode.ino ∈ files, links(old_parent_inode.ino, dirs) ≠ ∅ ∨ old_parent_inode.ino = ROOT_INO, links(new_parent_inode.ino, dirs) ≠ ∅ ∨ new_parent_inode.ino = ROOT_INO, old_child_inode.ino ∈ dirs ∧ old_child_inode.nlink = dirs[old_child_inode.ino].nlink ∧ old_child_inode.size = dirs[old_child_inode.ino].size ∧ old_child_inode.nsubdirs = dirs[old_child_inode.ino].nsubdirs ∧ old_child_inode.directory ∨ old_child_inode.ino ∈ files ∧ old_child_inode.nlink = files[old_child_inode.ino].nlink ∧ old_child_inode.nsubdirs = 0 ∧ ¬ old_child_inode.directory ⊦ ⟪cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, dirs, files, icache, dcache, asynced0)⟫ ( icache-cons(icache, dirs, files) ∧ dcache-cons(dcache, dirs, files) ∧ pcache-cons(pcache, dirs, files) ∧ afs-cons(dirs, files) ∧ synced(dirs, icache, pcache) ∧ (asynced → synced(files, icache, pcache))), str0 ∈ dirs[new_parent_inode.ino].entries

Rule argument

spec: dentry name: elim-02 seq: ⊦ dent.negdentry? → (str = dent.name ↔ dent = negdentry(str)) subst: [dent, str] → [new_dent, str0]

Additional information

Used simplifier rules: ⊦ dent.negdentry? → (str = dent.name ↔ dent = negdentry(str)); (elim) Used terms: new_dent, str0