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Proof step 62 from proof refine-rename-dir-negdentry (previous step)

# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries[str], cache_dirs) = cache_dirs[entries[str]].nlink, cache_dirs[m3].size = m0 + 1, cache_dirs[entries[str]] = mkdir(cache_dirs[entries[str]].size, md5, cache_dirs[entries[str]].nlink, cache_dirs[entries[str]].nsubdirs, entries2), cache_dirs[m] = mkdir(cache_dirs[m].size, md4, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), cache_dirs[m3] = mkdir(m0 + 1, md2, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries), entries[str] = cache_dirs[m3].entries[str], afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= (cache_dirs[m3].nsubdirs - 1)] [m, ( ( (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;cache_dirs[m] ) .entries:= (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;cache_dirs[m] ) .entries [str0, entries[str] ]) .size:= ( (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;cache_dirs[m] ) .size + 1)) .nsubdirs:= ( (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;cache_dirs[m] ) .nsubdirs + 1) ], cache_dirs[m3].nsubdirs ≠ 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), entries[str] ∈ cache_dirs, str ∈ entries, m3 ∈ cache_dirs, m ∈ cache_dirs, str ∈ cache_dirs[m3].entries, 1 ≤ cache_dirs[m3].nsubdirs, ¬ str0 ∈ cache_dirs[m].entries, ¬ str0 ∈ entries1, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_inode_sync#(mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, m0 + 1); dirs, cache_files); afs_inode_sync#(mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, cache_files); new_child_inode := new_child_inode0; skip; old_parent_inode := (m3 ∈ dirs ⊃ mkinode(m3, dirs[m3].meta, true, dirs[m3].nlink, dirs[m3].nsubdirs, dirs[m3].size) ;mkinode(m3, cache_files[m3].meta, false, cache_files[m3].nlink, 0, cache_files[m3].size) ); new_parent_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, cache_files[m].meta, false, cache_files[m].nlink, 0, cache_files[m].size) ); err := ESUCCESS; asynced := ?; new_dent := mkdentry(str0, mkdentry(str, entries[str]).ino); old_dent1 := negdentry(mkdentry(str, entries[str]).name); if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, cache_icache); icache_set#(new_parent_inode, asynced; err, cache_icache); skip; dcache_delete#(old_parent_inode.ino, old_dent1.name; err, cache_dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, cache_dcache)}] ( icache-cons(cache_icache, dirs, cache_files) ∧ dcache-cons(cache_dcache, dirs, cache_files) ∧ pcache-cons(cache_pcache, dirs, cache_files) ∧ afs-cons(dirs, cache_files) ∧ synced(dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); old_parent_inode0 := mkinode(m3, md0, true, # links(m3, cache_dirs), cache_dirs[m3].nsubdirs, m0 + 1); old_child_inode0 := mkinode(entries[str], md1, true, # links(entries[str], cache_dirs), cache_dirs[entries[str]].nsubdirs, cache_dirs[entries[str]].size); cache_dirs0 := cache_dirs; cache_dirs1 := cache_dirs; cache_dirs2 := cache_dirs; afs_rename# (; old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode0 ∧ new_parent_inode = new_parent_inode0 ∧ mkinode (entries[str], md1, true, # links(entries[str], cache_dirs0), cache_dirs1[entries[str]].nsubdirs, cache_dirs2[entries[str]].size) = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent1 = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 63:
# links(entries[str], cache_dirs) = cache_dirs[entries[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs = m1 + 1, cache_dirs[m3].size = m0 + 1, entries[str] = cache_dirs[m3].entries[str], cache_dirs[m3] = mkdir(m0 + 1, md2, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries), cache_dirs[m] = mkdir(cache_dirs[m].size, md4, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries1), cache_dirs[entries[str]] = mkdir(cache_dirs[entries[str]].size, md5, cache_dirs[entries[str]].nlink, cache_dirs[entries[str]].nsubdirs, entries2), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= m1] [m, ( ( (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= m1;cache_dirs[m]) .entries:= (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= m1;cache_dirs[m]) .entries [str0, entries[str] ]) .size:= ((m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= m1;cache_dirs[m]).size + 1)) .nsubdirs:= ((m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m0).nsubdirs:= m1;cache_dirs[m]).nsubdirs + 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[m3].entries, m ∈ cache_dirs, m3 ∈ cache_dirs, str ∈ entries, entries[str] ∈ cache_dirs, 1 ≤ cache_dirs[m3].nsubdirs, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ [afs_inode_sync#(mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, m0 + 1); dirs, cache_files); afs_inode_sync#(mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, cache_files); new_child_inode := new_child_inode0; skip; old_parent_inode := (m3 ∈ dirs ⊃ mkinode(m3, dirs[m3].meta, true, dirs[m3].nlink, dirs[m3].nsubdirs, dirs[m3].size) ;mkinode(m3, cache_files[m3].meta, false, cache_files[m3].nlink, 0, cache_files[m3].size) ); new_parent_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, cache_files[m].meta, false, cache_files[m].nlink, 0, cache_files[m].size) ); err := ESUCCESS; asynced := ?; new_dent := mkdentry(str0, mkdentry(str, entries[str]).ino); old_dent1 := negdentry(mkdentry(str, entries[str]).name); if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, cache_icache); icache_set#(new_parent_inode, asynced; err, cache_icache); skip; dcache_delete#(old_parent_inode.ino, old_dent1.name; err, cache_dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, cache_dcache)}] ( icache-cons(cache_icache, dirs, cache_files) ∧ dcache-cons(cache_dcache, dirs, cache_files) ∧ pcache-cons(cache_pcache, dirs, cache_files) ∧ afs-cons(dirs, cache_files) ∧ synced(dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨old_dent0 := mkdentry(str, entries[str]); new_dent0 := negdentry(str0); new_parent_inode0 := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); old_parent_inode0 := mkinode(m3, md0, true, # links(m3, cache_dirs), cache_dirs[m3].nsubdirs, m0 + 1); old_child_inode0 := mkinode(entries[str], md1, true, # links(entries[str], cache_dirs), cache_dirs[entries[str]].nsubdirs, cache_dirs[entries[str]].size); cache_dirs0 := cache_dirs; cache_dirs1 := cache_dirs; cache_dirs2 := cache_dirs; afs_rename# (; old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode0 ∧ new_parent_inode = new_parent_inode0 ∧ mkinode (entries[str], md1, true, # links(entries[str], cache_dirs0), cache_dirs1[entries[str]].nsubdirs, cache_dirs2[entries[str]].size) = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent1 = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced))), cache_dirs[m3].nsubdirs = 0, str0 ∈ cache_dirs[m].entries, str0 ∈ entries1

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [cache_dirs[m3].nsubdirs, m1]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: cache_dirs[m3].nsubdirs, m1