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Proof step 22 from proof inv-rmdir (previous step)

# links(entries[str], dirs0) = dirs0[entries[str]].nlink, # links(m, dirs0) = dirs0[m].nlink, dirs0[entries[str]].entries = ∅, entries[str] = dirs0[m].entries[str], dirs0[m] = mkdir(dirs0[m].size, md2, dirs0[m].nlink, dirs0[m].nsubdirs, entries), dirs0[entries[str]] = mkdir(dirs0[entries[str]].size, md1, dirs0[entries[str]].nlink, dirs0[entries[str]].nsubdirs, ∅), dirs = dirs0[m, ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1)).nsubdirs:= (dirs0[m].nsubdirs - 1)], dirs0[m].nsubdirs ≠ 0, dirs0[m].size ≠ 0, entries[str] ≠ ROOT_INO, afs-cons(dirs0, files), str ∈ dirs0[m].entries, m ∈ dirs0, str ∈ entries, entries[str] ∈ dirs0, 1 ≤ dirs0[m].nsubdirs, 1 ≤ dirs0[m].size, 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, md0, true, dirs0[m].nlink, dirs0[m].nsubdirs, dirs0[m].size); dirs, files); afs_inode_sync# (mkinode(entries[str], md, true, # links(entries[str], dirs0), dirs0[entries[str]].nsubdirs, dirs0[entries[str]].size); 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 := getinode(entries[str], dirs, files); err := ESUCCESS; asynced := ?⟫ afs-cons(dirs, files)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 23:
# links(m, dirs0) = m1, # links(entries[str], dirs0) = dirs0[entries[str]].nlink, dirs0[entries[str]].entries = ∅, dirs0[entries[str]] = mkdir(dirs0[entries[str]].size, md1, dirs0[entries[str]].nlink, dirs0[entries[str]].nsubdirs, ∅), dirs0[m] = mkdir(m0, md2, m1, m2, entries), entries[str] = entries0[str], dirs0[m] = mkdir(m0, md3, m1, m2, entries0), dirs = dirs0[m, ((dirs0[m].entries:= (entries0 -- str)).size:= (m0 - 1)).nsubdirs:= (m2 - 1)], afs-cons(dirs0, files), entries[str] ∈ dirs0, str ∈ entries, m ∈ dirs0, str ∈ entries0, 1 ≤ m0, 1 ≤ m2, 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, md0, true, m1, m2, m0); dirs, files); afs_inode_sync# (mkinode(entries[str], md, true, # links(entries[str], dirs0), dirs0[entries[str]].nsubdirs, dirs0[entries[str]].size); 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 := getinode(entries[str], dirs, files); err := ESUCCESS; asynced := ?⟫ afs-cons(dirs, files), entries[str] = ROOT_INO, m0 = 0, m2 = 0

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], 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: dirs0[m], m0, md3, m1, m2, entries0