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Proof step 806 from proof refine-rename-file-dentry (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, asynced0 ↔ old_ino = new_ino ∧ mkdentry(str, entries[str]) = mkdentry(str0, entries0[str0]), asynced1 ↔ true, entries0[str0] = cache_dirs[m3].entries[str0], entries[str] = cache_dirs[m].entries[str], cache_files[entries0[str0]] = mkfile(m0, md6, # links(entries0[str0], cache_dirs), content3), cache_dirs[m] = mkdir(cache_dirs[m].size, md7, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str0]] = mkfile(m8, md4, # links(entries0[str0], cache_dirs), content0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m12, md3, # links(entries[str], cache_dirs), content), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md8, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries0), cache_files[entries[str]] = mkfile(m1, md5, # links(entries[str], cache_dirs), content2), ¬ asynced0, dirs = cache_dirs[old_ino, cache_dirs[old_ino].entries:= (cache_dirs[old_ino].entries -- mkdentry(str0, entries0[str0]).name)], new_ino = m, old_dent = mkdentry(str0, entries0[str0]), old_ino = m3, 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), str0 ∈ cache_dirs[m3].entries, str ∈ cache_dirs[m].entries, m ∈ cache_dirs, m3 ∈ cache_dirs, entries[str] ∈ cache_files, entries0[str0] ∈ cache_files, str ∈ entries, str0 ∈ entries0, ⌊err0⌋ , old_ino = new_ino → str ∈ entries ∧ str0 ∈ entries0, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ (err0 = err ∧ (asynced ↔ asynced2)), entries[str] ∈ cache_dirs, dirs[old_ino].size ≥ 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 807:
# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[m3].size = 0, cache_files[entries[str]] = mkfile(m1, md5, # links(entries[str], cache_dirs), content2), cache_dirs[m3] = mkdir(0, md8, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m12, md3, # links(entries[str], cache_dirs), content), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str0]] = mkfile(m8, md4, # links(entries0[str0], cache_dirs), content0), cache_dirs[m] = mkdir(cache_dirs[m].size, md7, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries), cache_files[entries0[str0]] = mkfile(m0, md6, # links(entries0[str0], cache_dirs), content3), entries[str] = cache_dirs[m].entries[str], entries0[str0] = cache_dirs[m3].entries[str0], 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), str0 ∈ entries0, str ∈ entries, entries0[str0] ∈ cache_files, entries[str] ∈ cache_files, m3 ∈ cache_dirs, m ∈ cache_dirs, str ∈ cache_dirs[m].entries, str0 ∈ cache_dirs[m3].entries, ⌊err0⌋ , ¬ entries[str] ∈ cache_dirs, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ (err0 = err ∧ (asynced ↔ asynced2)), m3 = m ∧ mkdentry(str, entries[str]) = mkdentry(str0, entries0[str0])

Rule argument

Additional information

Used simplifier rules: ⊦ m < 1 ↔ m = 0; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ 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)