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Proof step 26 from proof crash-neutral-rmdir (previous step)

# links(m, dirs) = dirs[m].nlink, dirs[entries[str]] = mkdir(m3, md1, # links(entries[str], dirs), m2, ∅), dirs[m] = mkdir(dirs[m].size, md0, dirs[m].nlink, dirs[m].nsubdirs, entries), entries[str] = dirs[m].entries[str], dirs1 = dirs[m, ((dirs[m].entries:= (dirs[m].entries -- str)).size:= (dirs[m].size - 1)).nsubdirs:= (dirs[m].nsubdirs - 1)], files0 = files, dirs[m].nsubdirs ≠ 0, dirs[m].size ≠ 0, entries[str] ≠ ROOT_INO, afs-cons(dirs, files), str ∈ entries, m ∈ dirs, entries[str] ∈ dirs, str ∈ dirs[m].entries, 1 ≤ dirs[m].size, 1 ≤ dirs[m].nsubdirs, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ asynced, ⟨afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs1, files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(m, md, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size); dirs1, files); afs_inode_sync#(mkinode(entries[str], md2, true, # links(entries[str], dirs), m2, m3); dirs1, files); p_inode := (m ∈ dirs1 ⊃ mkinode(m, dirs1[m].meta, true, dirs1[m].nlink, dirs1[m].nsubdirs, dirs1[m].size) ;mkinode(m, files[m].meta, false, files[m].nlink, 0, files[m].size) ); c_inode := (entries[str] ∈ dirs1 ⊃ mkinode(entries[str], dirs1[entries[str]].meta, true, dirs1[entries[str]].nlink, dirs1[entries[str]].nsubdirs, dirs1[entries[str]].size) ;mkinode(entries[str], files[entries[str]].meta, false, files[entries[str]].nlink, 0, files[entries[str]].size) ); err := ESUCCESS; asynced := ?⟩ (asynced ∧ dirs \ orphans(dirs, files0) = dirs1 \ orphans(dirs1, files) ∧ files0 \ orphans(dirs, files0) = files \ orphans(dirs1, files))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 27:
# links(m, dirs) = m1, entries[str] = entries0[str], dirs[m] = mkdir(m0, md0, m1, m4, entries), dirs[entries[str]] = mkdir(m3, md1, # links(entries[str], dirs), m2, ∅), dirs[m] = mkdir(m0, md3, m1, m4, entries0), dirs1 = dirs[m, ((dirs[m].entries:= (entries0 -- str)).size:= (m0 - 1)).nsubdirs:= (m4 - 1)], files0 = files, afs-cons(dirs, files), str ∈ entries0, entries[str] ∈ dirs, m ∈ dirs, str ∈ entries, 1 ≤ m4, 1 ≤ m0, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ asynced, ⟨afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs1, files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(m, md, true, m1, m4, m0); dirs1, files); afs_inode_sync#(mkinode(entries[str], md2, true, # links(entries[str], dirs), m2, m3); dirs1, files); p_inode := (m ∈ dirs1 ⊃ mkinode(m, dirs1[m].meta, true, dirs1[m].nlink, dirs1[m].nsubdirs, dirs1[m].size) ;mkinode(m, files[m].meta, false, files[m].nlink, 0, files[m].size) ); c_inode := (entries[str] ∈ dirs1 ⊃ mkinode(entries[str], dirs1[entries[str]].meta, true, dirs1[entries[str]].nlink, dirs1[entries[str]].nsubdirs, dirs1[entries[str]].size) ;mkinode(entries[str], files[entries[str]].meta, false, files[entries[str]].nlink, 0, files[entries[str]].size) ); err := ESUCCESS; asynced := ?⟩ (asynced ∧ dirs \ orphans(dirs, files0) = dirs1 \ orphans(dirs1, files) ∧ files0 \ orphans(dirs, files0) = files \ orphans(dirs1, files)), entries[str] = ROOT_INO, m0 = 0, m4 = 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] → [dirs[m], m0, md, m1, m4, 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[m], m0, md, m1, m4, entries0