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Proof step 330 from proof refine-rmdir (previous step)

# links(m, cache_dirs) = m3, entries0[str] = entries[str], cache_dirs[entries0[str]] = mkdir(m0, md, # links(entries0[str], cache_dirs), m2, ∅), cache_dirs[m] = mkdir(m1, md1, m3, 0, entries0), cache_dirs[m] = mkdir(m1, md0, m3, m4, entries), m4 = 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 ∈ entries, str ∈ entries0, m ∈ cache_dirs, entries0[str] ∈ cache_dirs, 1 ≤ m1, ⌊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, m1 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 331:
cache_dirs[m] = mkdir(m1, md0, # links(m, cache_dirs), 0, entries), cache_dirs[entries[str]] = mkdir(m0, md, # links(entries[str], cache_dirs), m2, ∅), entries[str] ≠ ROOT_INO, m1 ≠ 0, 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), entries[str] ∈ cache_dirs, m ∈ cache_dirs, str ∈ entries, 1 ≤ m1, ⌊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: ⊦ 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