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Proof step 443 from proof refine-rename-dir-negdentry (previous step)

# links(entries1[str], cache_dirs) = cache_dirs[entries1[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, asynced0 ↔ false, asynced1 ↔ false, entries1[str] = cache_dirs[m3].entries[str], cache_dirs[m3] = mkdir(cache_dirs[m3].size, md4, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), cache_dirs[m] = mkdir(cache_dirs[m].size, md3, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_dirs[entries1[str]] = mkdir(cache_dirs[entries1[str]].size, md5, cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].nsubdirs, entries2), dirs = dirs0[old_ino, dirs0[old_ino].size:= (dirs0[old_ino].size - 1)], dirs0 = cache_dirs[old_ino, cache_dirs[old_ino].entries:= (cache_dirs[old_ino].entries -- mkdentry(str, entries1[str]).name)], new_ino = m, old_dent = mkdentry(str, entries1[str]), old_ino = m3, old_dent.dentry?, 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), str ∈ cache_dirs[m3].entries, m ∈ cache_dirs, m3 ∈ cache_dirs, entries1[str] ∈ cache_dirs, str ∈ entries1, old_dent.ino ∈ dirs, dirs0[old_ino].size ≥ 1, ⌊err0⌋ , m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ (err0 = err ∧ (asynced ↔ asynced2)), str0 ∈ entries0, str0 ∈ cache_dirs[m].entries, dirs[old_ino].nsubdirs ≥ 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 444:
# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries1[str], cache_dirs) = cache_dirs[entries1[str]].nlink, cache_dirs[m3].nsubdirs = 0, cache_dirs[entries1[str]] = mkdir(cache_dirs[entries1[str]].size, md5, cache_dirs[entries1[str]].nlink, cache_dirs[entries1[str]].nsubdirs, entries2), cache_dirs[m] = mkdir(cache_dirs[m].size, md3, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md4, cache_dirs[m3].nlink, 0, entries1), entries1[str] = cache_dirs[m3].entries[str], cache_dirs[m3].size ≠ 0, 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 ∈ entries1, m3 ∈ cache_dirs, m ∈ cache_dirs, str ∈ cache_dirs[m3].entries, entries1[str] ∈ cache_dirs, 1 ≤ cache_dirs[m3].size, ⌊err0⌋ , ¬ str0 ∈ cache_dirs[m].entries, ¬ str0 ∈ entries0, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ (err0 = err ∧ (asynced ↔ asynced2))

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)% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (s)% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)% ⊦ mkdentry(str, m).ino = m; (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)