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Proof step 19 from proof inv-unlink (previous step)

# links(m, dirs0) = dirs0[m].nlink, # links(entries[str], dirs0) = c_inode.nlink, c_inode.ino = entries[str], c_inode.nsubdirs = 0, files[entries[str]] = mkfile(m0, md1, c_inode.nlink, content), entries[str] = dirs0[m].entries[str], dirs0[m] = mkdir(dirs0[m].size, md0, dirs0[m].nlink, dirs0[m].nsubdirs, entries), dirs = dirs0[m, (dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1)], dirs0[m].size ≠ 0, afs-cons(dirs0, files), str ∈ dirs0[m].entries, m ∈ dirs0, str ∈ entries, entries[str] ∈ files, 1 ≤ dirs0[m].size, ¬ c_inode.directory, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(m, md, true, dirs0[m].nlink, dirs0[m].nsubdirs, dirs0[m].size); dirs, files); afs_inode_sync#(c_inode; dirs, files); p_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, files[m].meta, false, files[m].nlink, 0, files[m].size) ); c_inode := (c_inode.ino ∈ dirs ⊃ mkinode(c_inode.ino, dirs[c_inode.ino].meta, true, dirs[c_inode.ino].nlink, dirs[c_inode.ino].nsubdirs, dirs[c_inode.ino].size) ;mkinode(c_inode.ino, files[c_inode.ino].meta, false, files[c_inode.ino].nlink, 0, files[c_inode.ino].size) ); err := ESUCCESS; asynced := ?⟫ afs-cons(dirs, files)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 20:
# links(entries[str], dirs0) = c_inode.nlink, # links(m, dirs0) = m2, c_inode.ino = entries[str], c_inode.nsubdirs = 0, dirs0[m] = mkdir(m1, md0, m2, m3, entries), entries[str] = entries0[str], files[entries[str]] = mkfile(m0, md1, c_inode.nlink, content), dirs0[m] = mkdir(m1, md2, m2, m3, entries0), dirs = dirs0[m, (dirs0[m].entries:= (entries0 -- str)).size:= (m1 - 1)], afs-cons(dirs0, files), entries[str] ∈ files, str ∈ entries, m ∈ dirs0, str ∈ entries0, 1 ≤ m1, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(m, md, true, m2, m3, m1); dirs, files); afs_inode_sync#(c_inode; dirs, files); p_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, files[m].meta, false, files[m].nlink, 0, files[m].size) ); c_inode := (c_inode.ino ∈ dirs ⊃ mkinode(c_inode.ino, dirs[c_inode.ino].meta, true, dirs[c_inode.ino].nlink, dirs[c_inode.ino].nsubdirs, dirs[c_inode.ino].size) ;mkinode(c_inode.ino, files[c_inode.ino].meta, false, files[c_inode.ino].nlink, 0, files[c_inode.ino].size) ); err := ESUCCESS; asynced := ?⟫ afs-cons(dirs, files), m1 = 0, c_inode.directory

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] → [dirs0[m], m1, md0, m2, m3, 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: dirs0[m], m1, md0, m2, m3, entries0