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Proof step 16 from proof inv-rmdir (previous step)

# links(entries[str], dirs) = dirs[entries[str]].nlink, # links(m, dirs) = dirs[m].nlink, dirs[entries[str]].entries = ∅, entries[str] = dirs[m].entries[str], dirs[entries[str]] = mkdir(dirs[entries[str]].size, md1, dirs[entries[str]].nlink, dirs[entries[str]].nsubdirs, ∅), dirs[m] = mkdir(dirs[m].size, md2, dirs[m].nlink, dirs[m].nsubdirs, entries), c_inode = mkinode(entries[str], md0, true, dirs[entries[str]].nlink, dirs[entries[str]].nsubdirs, dirs[entries[str]].size), dent = mkdentry(str, entries[str]), p_inode = mkinode(m, md, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size), entries[str] ≠ ROOT_INO, 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[m].entries, entries[str] ∈ dirs, str ∈ entries, m ∈ dirs, asynced → synced(files, icache, pcache), links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_rmdir#(; p_inode, c_inode, dent, err, dirs, files; asynced0); if err = ESUCCESS then let asynced = false in {icache_set#(c_inode, asynced; err, icache); icache_set#(p_inode, asynced; err, icache); dcache_delete#(p_inode.ino, dent.name; err, dcache)}⟫ ( 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 17:
# links(m, dirs) = dirs[m].nlink, # links(entries[str], dirs) = m1, dirs[m] = mkdir(dirs[m].size, md2, dirs[m].nlink, dirs[m].nsubdirs, entries), dirs[entries[str]] = mkdir(m0, md1, m1, m2, ∅), entries[str] = dirs[m].entries[str], dirs[entries[str]] = mkdir(m0, md3, m1, m2, entries0), c_inode = mkinode(entries[str], md0, true, m1, m2, m0), dent = mkdentry(str, entries[str]), entries0 = ∅, p_inode = mkinode(m, md, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size), afs-cons(dirs, files), dcache-cons(dcache, dirs, files), icache-cons(icache, dirs, files), pcache-cons(pcache, dirs, files), synced(dirs, icache, pcache), m ∈ dirs, str ∈ entries, entries[str] ∈ dirs, str ∈ dirs[m].entries, asynced → synced(files, icache, pcache), links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_rmdir#(; p_inode, c_inode, dent, err, dirs, files; asynced0); if err = ESUCCESS then let asynced = false in {icache_set#(c_inode, asynced; err, icache); icache_set#(p_inode, asynced; err, icache); dcache_delete#(p_inode.ino, dent.name; err, dcache)}⟫ ( 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)), entries[str] = ROOT_INO

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [dirs[entries[str]], m0, md3, m1, m2, entries0]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: dirs[entries[str]], m0, md3, m1, m2, entries0