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Proof step 38 from proof inv-rmdir (previous step)

# links(m, dirs0) = m1, # links(entries[str], dirs0) = dirs0[entries[str]].nlink, dirs0[entries[str]].entries = ∅, dirs0[m] = mkdir(m0, md3, m1, m2, entries), dirs0[entries[str]] = mkdir(dirs0[entries[str]].size, md2, dirs0[entries[str]].nlink, dirs0[entries[str]].nsubdirs, ∅), entries[str] = entries0[str], dirs0[m] = mkdir(m0, md1, m1, m2, entries0), dirs = dirs0[m, ((dirs0[m].entries:= (entries0 -- str)).size:= (m0 - 1)).nsubdirs:= (m2 - 1)], afs-cons(dirs0, files), dcache-cons(dcache, dirs0, files), icache-cons(icache, dirs0, files), pcache-cons(pcache, dirs0, files), synced(dirs0, icache, pcache), m ∈ dirs0, str ∈ entries, entries[str] ∈ dirs0, str ∈ entries0, 1 ≤ m0, 1 ≤ m2, asynced1 → synced(files, icache, pcache), 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, m1, m2, m0); dirs, files); afs_inode_sync# (mkinode(entries[str], md0, 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) → ⟪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) ∧ (asynced1 → synced(files, icache, pcache)))), entries[str] = ROOT_INO, m0 = 0, m2 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 39:
# links(entries0[str], dirs0) = dirs0[entries0[str]].nlink, dirs0[entries0[str]].entries = ∅, dirs0[entries0[str]] = mkdir(dirs0[entries0[str]].size, md2, dirs0[entries0[str]].nlink, dirs0[entries0[str]].nsubdirs, ∅), dirs0[m] = mkdir(m0, md1, # links(m, dirs0), m2, entries0), dirs = dirs0[m, mkdir(m0 - 1, md1, # links(m, dirs0), m2 - 1, entries0 -- str)], entries0[str] ≠ ROOT_INO, m0 ≠ 0, m2 ≠ 0, afs-cons(dirs0, files), dcache-cons(dcache, dirs0, files), icache-cons(icache, dirs0, files), pcache-cons(pcache, dirs0, files), synced(dirs0, icache, pcache), entries0[str] ∈ dirs0, str ∈ entries0, m ∈ dirs0, 1 ≤ m2, 1 ≤ m0, asynced1 → synced(files, icache, pcache), links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries0[str]).ino; dirs, files); dent := negdentry(mkdentry(str, entries0[str]).name); afs_inode_sync#(mkinode(m, md, true, # links(m, dirs0), m2, m0); dirs, files); afs_inode_sync# (mkinode(entries0[str], md0, true, # links(entries0[str], dirs0), dirs0[entries0[str]].nsubdirs, dirs0[entries0[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(entries0[str], dirs, files); err := ESUCCESS; asynced := ?] ( afs-cons(dirs, files) → ⟪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) ∧ (asynced1 → synced(files, icache, pcache))))

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%afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base