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Proof step 326 from proof refine-rmdir (previous step)

# links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[entries0[str]].entries = ∅, entries0[str] = cache_dirs[m].entries[str], cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md3, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), cache_dirs[m] = mkdir(cache_dirs[m].size, md1, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), p_ino = m, afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), str ∈ cache_dirs[m].entries, str ∈ entries0, m ∈ cache_dirs, entries0[str] ∈ cache_dirs, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size ≥ 1, ⌊err0⌋ , cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ ((asynced ↔ asynced0) ∧ err0 = err), entries0[str] = ROOT_INO, ( str ∈ entries0 ∧ str ∈ entries0 ∧ cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size ≥ 1 ∧ str ∈ entries0) ∧ ( str ∈ entries0 ∧ str ∈ entries0 ∧ cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size ≥ 1 ∧ str ∈ entries0) ∧ cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino] .size:= (cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size - 1) ] [p_ino ] .nsubdirs ≥ 1 ∧ str ∈ entries0 ∧ str ∈ entries0 ∧ cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries0[str]).name)][p_ino].size ≥ 1 ∧ str ∈ entries0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 327:
# links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].entries = ∅, cache_dirs[m].nsubdirs = 0, cache_dirs[m] = mkdir(cache_dirs[m].size, md1, cache_dirs[m].nlink, 0, entries0), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md3, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), entries0[str] = cache_dirs[m].entries[str], cache_dirs[m].size ≠ 0, entries0[str] ≠ ROOT_INO, afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), entries0[str] ∈ cache_dirs, m ∈ cache_dirs, str ∈ entries0, str ∈ cache_dirs[m].entries, 1 ≤ cache_dirs[m].size, ⌊err0⌋ , cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ ((asynced ↔ asynced0) ∧ err0 = err)

Rule argument

Additional information

Used simplifier rules: not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat% ⊦ m < 1 ↔ m = 0; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ (ad.entries:= entries).nsubdirs = ad.nsubdirs; (s)% ⊦ (ad.size:= m).nsubdirs = ad.nsubdirs; (s)% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ (ad.entries:= entries).size = ad.size; (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)% ⊦ mkdentry(str, m).name = str; (s)