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Proof step 27 from proof crash-neutral-rmdir (previous step)

# 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 28:
dirs[entries0[str]] = mkdir(m3, md1, # links(entries0[str], dirs), m2, ∅), dirs[m] = mkdir(m0, md3, # links(m, dirs), m4, entries0), dirs1 = dirs[m, mkdir(m0 - 1, md3, # links(m, dirs), m4 - 1, entries0 -- str)], files0 = files, entries0[str] ≠ ROOT_INO, m0 ≠ 0, m4 ≠ 0, afs-cons(dirs, files), m ∈ dirs, entries0[str] ∈ dirs, str ∈ entries0, 1 ≤ m0, 1 ≤ m4, links(m, dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟨afs_drop_nlink#(mkdentry(str, entries0[str]).ino; dirs1, files); dent := negdentry(mkdentry(str, entries0[str]).name); afs_inode_sync#(mkinode(m, md, true, # links(m, dirs), m4, m0); dirs1, files); afs_inode_sync#(mkinode(entries0[str], md2, true, # links(entries0[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 := (entries0[str] ∈ dirs1 ⊃ mkinode(entries0[str], dirs1[entries0[str]].meta, true, dirs1[entries0[str]].nlink, dirs1[entries0[str]].nsubdirs, dirs1[entries0[str]].size) ;mkinode(entries0[str], files[entries0[str]].meta, false, files[entries0[str]].nlink, 0, files[entries0[str]].size) ); err := ESUCCESS; asynced := ?⟩ (asynced ∧ dirs \ orphans(dirs, files0) = dirs1 \ orphans(dirs1, files) ∧ files0 \ orphans(dirs, files0) = files \ orphans(dirs1, files)), ⟨fail#(; ; err); asynced := ?⟩ asynced

Rule argument

Additional information

Used simplifier rules: ⊦ mkdir(m, md, m0, m1, entries).nsubdirs:= m4 = mkdir(m, md, m0, m4, entries); (s)%⊦ mkdir(m, md, m0, m1, entries).size:= m2 = mkdir(m2, md, m0, m1, entries); (s)%⊦ mkdir(m, md, m0, m1, entries).entries:= entries0 = mkdir(m, md, m0, m1, entries0); (s)%⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat