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Proof step 289 from proof refine-link (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[ino1] = mkdir(m7, md5, # links(ino1, cache_dirs), m11, entries1), entries[str] = entries1[str], cache_dirs[ino0] = mkdir(m4, md2, # links(ino0, cache_dirs), m10, entries), cache_dirs[ino3] = mkdir(m0, md3, # links(ino3, cache_dirs), m2, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries1[str]] = mkfile(m5, md1, # links(entries1[str], cache_dirs), content), cache_dirs[m] = mkdir(cache_dirs[m].size, md4, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries2), entries0[str] = entries1[str], cache_files[entries1[str]] = mkfile(m3, md6, # links(entries1[str], cache_dirs), content1), entries1[str] = cache_dirs[ino2].entries[str], afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, (cache_dirs[m].entries:= cache_dirs[m].entries[str0, entries1[str]]).size:= (cache_dirs[m].size + 1)], 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 ∈ entries0, str ∈ entries1, ino1 ∈ cache_dirs, entries1[str] ∈ cache_files, m ∈ cache_dirs, ino3 ∈ cache_dirs, ino0 ∈ cache_dirs, str ∈ entries, ino2 ∈ cache_dirs, str ∈ cache_dirs[ino2].entries, ⌊err0⌋ , ¬ str0 ∈ cache_dirs[m].entries, ¬ str0 ∈ entries2, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(ino1, cache_dirs) ≠ ∅ ∨ ino1 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(ino3, cache_dirs) ≠ ∅ ∨ ino3 = ROOT_INO, links(ino0, cache_dirs) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino2, cache_dirs) ≠ ∅ ∨ ino2 = ROOT_INO ⊦ ⟨afs_inc_nlink#(mkdentry(str, entries0[str]).ino; dirs, afs_files); new_dent := mkdentry(str0, mkdentry(str, entries0[str]).ino); afs_inode_sync#(mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, afs_files); afs_inode_sync#(mkinode(entries1[str], md0, false, # links(entries1[str], cache_dirs), 0, m6); dirs, afs_files); p_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, afs_files[m].meta, false, afs_files[m].nlink, 0, afs_files[m].size) ); c_inode := getinode(mkdentry(str, entries0[str]).ino, dirs, afs_files); err := ESUCCESS; asynced := ?⟩ ( dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (asynced ↔ asynced0) ∧ mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size) = p_inode ∧ mkinode(entries1[str], md0, false, # links(entries1[str], cache_dirs), 0, m6) = c_inode ∧ negdentry(str0) = new_dent ∧ err0 = err), ⟨fail#(; ; err); asynced := ?⟩ ((asynced ↔ asynced0) ∧ err0 = err)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 290:
# links(m, cache_dirs) = m8, entries1[str] = cache_dirs[ino2].entries[str], cache_files[entries1[str]] = mkfile(m3, md6, # links(entries1[str], cache_dirs), content1), entries0[str] = entries1[str], cache_dirs[m] = mkdir(m1, md4, m8, m9, entries2), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries1[str]] = mkfile(m5, md1, # links(entries1[str], cache_dirs), content), cache_dirs[ino3] = mkdir(m0, md3, # links(ino3, cache_dirs), m2, entries0), cache_dirs[ino0] = mkdir(m4, md2, # links(ino0, cache_dirs), m10, entries), entries[str] = entries1[str], cache_dirs[ino1] = mkdir(m7, md5, # links(ino1, cache_dirs), m11, entries1), cache_dirs[m] = mkdir(m1, md7, m8, m9, entries3), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, (cache_dirs[m].entries:= entries3[str0, entries1[str]]).size:= (m1 + 1)], 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[ino2].entries, ino2 ∈ cache_dirs, str ∈ entries, ino0 ∈ cache_dirs, ino3 ∈ cache_dirs, m ∈ cache_dirs, entries1[str] ∈ cache_files, ino1 ∈ cache_dirs, str ∈ entries1, str ∈ entries0, ⌊err0⌋ , cache_sync → synced(cache_files, cache_icache, cache_pcache), links(ino2, cache_dirs) ≠ ∅ ∨ ino2 = ROOT_INO, links(ino0, cache_dirs) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino3, cache_dirs) ≠ ∅ ∨ ino3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(ino1, cache_dirs) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ ⟨afs_inc_nlink#(mkdentry(str, entries0[str]).ino; dirs, afs_files); new_dent := mkdentry(str0, mkdentry(str, entries0[str]).ino); afs_inode_sync#(mkinode(m, md, true, m8, m9, m1); dirs, afs_files); afs_inode_sync#(mkinode(entries1[str], md0, false, # links(entries1[str], cache_dirs), 0, m6); dirs, afs_files); p_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, afs_files[m].meta, false, afs_files[m].nlink, 0, afs_files[m].size) ); c_inode := getinode(mkdentry(str, entries0[str]).ino, dirs, afs_files); err := ESUCCESS; asynced := ?⟩ ( dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (asynced ↔ asynced0) ∧ mkinode(m, md, true, # links(m, cache_dirs), m9, m1) = p_inode ∧ mkinode(entries1[str], md0, false, # links(entries1[str], cache_dirs), 0, m6) = c_inode ∧ negdentry(str0) = new_dent ∧ err0 = err), ⟨fail#(; ; err); asynced := ?⟩ ((asynced ↔ asynced0) ∧ err0 = err), str0 ∈ entries3, str0 ∈ entries2

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[m], m1, md7, m8, m9, entries3]

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[m], m1, md7, m8, m9, entries3