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Proof step 112 from proof refine-link (previous step)

# links(m9, cache_dirs) = m1, entries1[str] = cache_dirs[ino3].entries[str], cache_dirs[ino2] = mkdir(m7, md5, # links(ino2, cache_dirs), m11, entries1), entries[str] = entries1[str], cache_files[entries1[str]] = mkfile(m3, md6, # links(entries1[str], cache_dirs), content1), entries0[str] = entries1[str], cache_dirs[m9] = mkdir(m, md4, m1, m6, entries2), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries1[str]] = mkfile(m5, md1, # links(entries1[str], cache_dirs), content), cache_dirs[ino1] = mkdir(m0, md3, # links(ino1, cache_dirs), m2, entries0), cache_dirs[ino0] = mkdir(m4, md2, # links(ino0, cache_dirs), m10, entries), cache_dirs[m9] = mkdir(m, md7, m1, m6, entries3), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m9, (cache_dirs[m9].entries:= entries3[str0, entries1[str]]).size:= (m + 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[ino3].entries, ino3 ∈ cache_dirs, entries1[str] ∈ cache_files, m9 ∈ cache_dirs, str ∈ entries1, ino2 ∈ cache_dirs, ino0 ∈ cache_dirs, ino1 ∈ cache_dirs, str ∈ entries, str ∈ entries0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(ino3, cache_dirs) ≠ ∅ ∨ ino3 = ROOT_INO, links(ino2, cache_dirs) ≠ ∅ ∨ ino2 = ROOT_INO, links(ino0, cache_dirs) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino1, cache_dirs) ≠ ∅ ∨ ino1 = ROOT_INO, links(m9, cache_dirs) ≠ ∅ ∨ m9 = 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(m9, md0, true, m1, m6, m); dirs, afs_files); afs_inode_sync#(mkinode(entries1[str], md, false, # links(entries1[str], cache_dirs), 0, m14); dirs, afs_files); p_inode := (m9 ∈ dirs ⊃ mkinode(m9, dirs[m9].meta, true, dirs[m9].nlink, dirs[m9].nsubdirs, dirs[m9].size) ;mkinode(m9, afs_files[m9].meta, false, afs_files[m9].nlink, 0, afs_files[m9].size) ); c_inode := getinode(mkdentry(str, entries0[str]).ino, dirs, afs_files); err := ESUCCESS; asynced := ?⟩ ( dirs = cache_dirs[m9, mkdir(m + 1, md0, # links(m9, cache_dirs), m6, entries2[str0, entries1[str]])] ∧ afs_files = ( cache_files[entries1[str], mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1)] [entries[str], mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ] ⊕ cache_pcache) ⊕ cache_icache[m9, mkientry(false, mkinode(m9, md0, true, # links(m9, cache_dirs), m6, m + 1))] [entries1[str], mkientry (false, (entries0[str] ∈ cache_dirs ⊃ mkinode (entries0[str], (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries2[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .meta, true, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries2[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .nlink, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries2[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .nsubdirs, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries2[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .size) ;mkinode (entries0[str], (entries0[str] = entries[str] ∧ entries[str] ∈ cache_files ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .meta, false, (entries0[str] = entries[str] ∧ entries[str] ∈ cache_files ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .nlink, 0, (entries0[str] = entries[str] ∧ entries[str] ∈ cache_files ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .size) )) ] ∧ (asynced ↔ asynced0) ∧ mkinode(m9, md0, true, # links(m9, cache_dirs), m6, m + 1) = p_inode ∧ (entries0[str] ∈ cache_dirs ⊃ mkinode (entries0[str], (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries2[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .meta, true, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries2[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .nlink, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries2[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .nsubdirs, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries2[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .size) ;mkinode (entries0[str], (entries0[str] = entries[str] ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .meta, false, (entries0[str] = entries[str] ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .nlink, 0, (entries0[str] = entries[str] ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .size) ) = c_inode ∧ mkdentry(str0, entries0[str]) = new_dent ∧ ESUCCESS = err), str0 ∈ entries3, str0 ∈ entries2

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 113:
cache_dirs[ino0] = mkdir(m4, md2, # links(ino0, cache_dirs), m10, entries), cache_dirs[ino1] = mkdir(m0, md3, # links(ino1, cache_dirs), m2, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries1[str]] = mkfile(m5, md1, # links(entries1[str], cache_dirs), content), cache_dirs[m9] = mkdir(m, md7, # links(m9, cache_dirs), m6, entries3), entries0[str] = entries1[str], cache_files[entries1[str]] = mkfile(m3, md6, # links(entries1[str], cache_dirs), content1), entries[str] = entries1[str], cache_dirs[ino2] = mkdir(m7, md5, # links(ino2, cache_dirs), m11, entries1), entries1[str] = cache_dirs[ino3].entries[str], afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m9, mkdir(m + 1, md7, # links(m9, cache_dirs), m6, entries3[str0, entries1[str]])], 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 ∈ entries, ino1 ∈ cache_dirs, ino0 ∈ cache_dirs, ino2 ∈ cache_dirs, str ∈ entries1, m9 ∈ cache_dirs, entries1[str] ∈ cache_files, ino3 ∈ cache_dirs, str ∈ cache_dirs[ino3].entries, ¬ str0 ∈ entries3, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m9, cache_dirs) ≠ ∅ ∨ m9 = ROOT_INO, links(ino1, cache_dirs) ≠ ∅ ∨ ino1 = ROOT_INO, links(ino0, cache_dirs) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino2, cache_dirs) ≠ ∅ ∨ ino2 = ROOT_INO, links(ino3, cache_dirs) ≠ ∅ ∨ ino3 = 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(m9, md0, true, # links(m9, cache_dirs), m6, m); dirs, afs_files); afs_inode_sync#(mkinode(entries1[str], md, false, # links(entries1[str], cache_dirs), 0, m14); dirs, afs_files); p_inode := (m9 ∈ dirs ⊃ mkinode(m9, dirs[m9].meta, true, dirs[m9].nlink, dirs[m9].nsubdirs, dirs[m9].size) ;mkinode(m9, afs_files[m9].meta, false, afs_files[m9].nlink, 0, afs_files[m9].size) ); c_inode := getinode(mkdentry(str, entries0[str]).ino, dirs, afs_files); err := ESUCCESS; asynced := ?⟩ ( dirs = cache_dirs[m9, mkdir(m + 1, md0, # links(m9, cache_dirs), m6, entries3[str0, entries1[str]])] ∧ afs_files = ( cache_files[entries1[str], mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1)] [entries[str], mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ] ⊕ cache_pcache) ⊕ cache_icache[m9, mkientry(false, mkinode(m9, md0, true, # links(m9, cache_dirs), m6, m + 1))] [entries1[str], mkientry (false, (entries0[str] ∈ cache_dirs ⊃ mkinode (entries0[str], (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries3[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .meta, true, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries3[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .nlink, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries3[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .nsubdirs, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries3[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .size) ;mkinode (entries0[str], (entries0[str] = entries[str] ∧ entries[str] ∈ cache_files ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .meta, false, (entries0[str] = entries[str] ∧ entries[str] ∈ cache_files ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .nlink, 0, (entries0[str] = entries[str] ∧ entries[str] ∈ cache_files ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .size) )) ] ∧ (asynced ↔ asynced0) ∧ mkinode(m9, md0, true, # links(m9, cache_dirs), m6, m + 1) = p_inode ∧ (entries0[str] ∈ cache_dirs ⊃ mkinode (entries0[str], (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries3[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .meta, true, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries3[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .nlink, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries3[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .nsubdirs, (entries0[str] = m9 ⊃ mkdir (cache_dirs[entries0[str]].size + 1, md0, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, entries3[str0, entries1[str]]) ;cache_dirs[entries0[str]] ) .size) ;mkinode (entries0[str], (entries0[str] = entries[str] ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .meta, false, (entries0[str] = entries[str] ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .nlink, 0, (entries0[str] = entries[str] ⊃ mkfile(m14, md, # links(entries1[str], cache_dirs) + 1, content1) ;mkfile(m3, md6, # links(entries1[str], cache_dirs) + 1, content1) ) .size) ) = c_inode ∧ mkdentry(str0, entries0[str]) = new_dent ∧ ESUCCESS = err)

Rule argument

Additional information

Used simplifier rules: ⊦ mkdir(m, md, m0, m1, entries).size:= m2 = mkdir(m2, md, m0, m1, entries); (s)%⊦ mkdir(m, md, m0, m1, entries).entries:= entries0 = mkdir(m, md, m0, m1, entries0); (s)%⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)