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Proof step 752 from proof refine-rename-dir-dentry (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 = ∅, cache_dirs[m3] = mkdir(0, md1, # links(m3, cache_dirs), m2, entries), 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[entries[str0]] = mkdir(m0, md2, # links(entries[str0], cache_dirs), m4, entries1), entries0[str] = cache_dirs[m].entries[str], 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), str ∈ entries0, m3 ∈ cache_dirs, m ∈ cache_dirs, entries[str0] ∈ cache_dirs, entries0[str] ∈ cache_dirs, str ∈ cache_dirs[m].entries, str0 ∈ entries, ⌊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, entries[str0])

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 753:
# links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries0[str], cache_dirs) = m5, entries0[str] = cache_dirs[m].entries[str], cache_dirs[entries[str0]] = mkdir(m0, md2, # links(entries[str0], cache_dirs), m4, entries1), cache_dirs[entries0[str]] = mkdir(m1, md4, m5, m6, ∅), cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_dirs[m3] = mkdir(0, md1, # links(m3, cache_dirs), m2, entries), cache_dirs[entries0[str]] = mkdir(m1, md3, m5, m6, entries2), entries2 = ∅, 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 ∈ entries, str ∈ cache_dirs[m].entries, entries0[str] ∈ cache_dirs, entries[str0] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, str ∈ entries0, ⌊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)), entries0[str] = ROOT_INO, m3 = m ∧ mkdentry(str, entries0[str]) = mkdentry(str0, entries[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[entries0[str]], m1, md3, m5, m6, entries2]

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[entries0[str]], m1, md3, m5, m6, entries2