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Proof step 3 from proof inv-rmdir (previous step)

dirs[dent.ino].entries = ∅, c_inode.ino = dent.ino, dent.ino = dirs[p_inode.ino].entries[dent.name], c_inode.nlink = dirs[dent.ino].nlink, p_inode.nlink = dirs[p_inode.ino].nlink, c_inode.nsubdirs = dirs[dent.ino].nsubdirs, p_inode.nsubdirs = dirs[p_inode.ino].nsubdirs, c_inode.size = dirs[dent.ino].size, p_inode.size = dirs[p_inode.ino].size, dent.ino ≠ ROOT_INO, dent.dentry?, c_inode.directory, p_inode.directory, afs-cons(dirs, files), dcache-cons(dcache, dirs, files), icache-cons(icache, dirs, files), pcache-cons(pcache, dirs, files), synced(dirs, icache, pcache), dent.ino ∈ dirs, dent.name ∈ dirs[p_inode.ino].entries, p_inode.ino ∈ dirs, asynced → synced(files, icache, pcache), links(p_inode.ino, dirs) ≠ ∅ ∨ p_inode.ino = ROOT_INO ⊦ ⟪cache_rmdir#(; p_inode, c_inode, dent, err, dirs, files, icache, dcache, asynced0)⟫ ( icache-cons(icache, dirs, files) ∧ dcache-cons(dcache, dirs, files) ∧ pcache-cons(pcache, dirs, files) ∧ synced(dirs, icache, pcache) ∧ (asynced → synced(files, icache, pcache)) ∧ afs-cons(dirs, files))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
dirs[m].entries = ∅, c_inode.ino = m, p_inode.nlink = dirs[p_inode.ino].nlink, c_inode.nlink = dirs[m].nlink, p_inode.nsubdirs = dirs[p_inode.ino].nsubdirs, c_inode.nsubdirs = dirs[m].nsubdirs, p_inode.size = dirs[p_inode.ino].size, c_inode.size = dirs[m].size, dent = mkdentry(str, m), m = dirs[p_inode.ino].entries[str], dent.dentry?, p_inode.directory, c_inode.directory, afs-cons(dirs, files), dcache-cons(dcache, dirs, files), icache-cons(icache, dirs, files), pcache-cons(pcache, dirs, files), synced(dirs, icache, pcache), p_inode.ino ∈ dirs, str ∈ dirs[p_inode.ino].entries, m ∈ dirs, asynced → synced(files, icache, pcache), links(p_inode.ino, dirs) ≠ ∅ ∨ p_inode.ino = ROOT_INO ⊦ ⟪cache_rmdir#(; p_inode, c_inode, dent, err, dirs, files, icache, dcache, asynced0)⟫ ( icache-cons(icache, dirs, files) ∧ dcache-cons(dcache, dirs, files) ∧ pcache-cons(pcache, dirs, files) ∧ synced(dirs, icache, pcache) ∧ (asynced → synced(files, icache, pcache)) ∧ afs-cons(dirs, files)), m = ROOT_INO

Rule argument

spec: dentry name: elim seq: ⊦ dent.dentry? → (str = dent.name ∧ m = dent.ino ↔ dent = mkdentry(str, m)) subst: [dent, str, m] → [dent, str, m]

Additional information

Used simplifier rules: ⊦ dent.dentry? → (str = dent.name ∧ m = dent.ino ↔ dent = mkdentry(str, m)); (elim) Used terms: dent, str, m