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Proof step 723 from proof refine-rename-dir-dentry (previous step)

# links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries1[str0], cache_dirs) = cache_dirs[entries1[str0]].nlink, cache_dirs[entries0[str]].entries = ∅, cache_dirs[m3].nsubdirs = 0, cache_dirs[m3] = mkdir(cache_dirs[m3].size, md7, cache_dirs[m3].nlink, 0, entries1), cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md4, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), cache_dirs[entries1[str0]] = mkdir(cache_dirs[entries1[str0]].size, md3, cache_dirs[entries1[str0]].nlink, cache_dirs[entries1[str0]].nsubdirs, entries2), entries0[str] = cache_dirs[m].entries[str], entries1[str0] = cache_dirs[m3].entries[str0], cache_dirs[m3].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), str0 ∈ entries1, str ∈ entries0, m3 ∈ cache_dirs, m ∈ cache_dirs, entries0[str] ∈ cache_dirs, str ∈ cache_dirs[m].entries, str0 ∈ cache_dirs[m3].entries, entries1[str0] ∈ cache_dirs, 1 ≤ cache_dirs[m3].size, ⌊err0⌋ , 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, entries0[str]) = mkdentry(str0, entries1[str0])

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 724:
# links(entries1[str0], cache_dirs) = m1, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].entries = ∅, cache_dirs[m3].nsubdirs = 0, entries1[str0] = cache_dirs[m3].entries[str0], entries0[str] = cache_dirs[m].entries[str], cache_dirs[entries1[str0]] = mkdir(m0, md3, m1, m2, entries2), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md4, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md7, cache_dirs[m3].nlink, 0, entries1), cache_dirs[entries1[str0]] = mkdir(m0, md1, m1, m2, entries), 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), entries1[str0] ∈ cache_dirs, str0 ∈ cache_dirs[m3].entries, str ∈ cache_dirs[m].entries, entries0[str] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, str ∈ entries0, str0 ∈ entries1, 1 ≤ cache_dirs[m3].size, ⌊err0⌋ , 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)), cache_dirs[m3].size = 0, entries0[str] = ROOT_INO, m3 = m ∧ mkdentry(str, entries0[str]) = mkdentry(str0, entries1[str0])

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [cache_dirs[entries1[str0]], m0, md1, m1, m2, entries]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: cache_dirs[entries1[str0]], m0, md1, m1, m2, entries