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Proof step 64 from proof refine-rename-file-negdentry (previous step)

# links(entries1[str], cache_dirs) = cache_files[entries1[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, entries1[str] = cache_dirs[m3].entries[str], cache_dirs[m3] = mkdir(cache_dirs[m3].size, md6, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries1[str]] = mkfile(m9, md2, cache_files[entries1[str]].nlink, content), cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_files[entries1[str]] = mkfile(m0, md4, cache_files[entries1[str]].nlink, content1), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, (cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= (cache_dirs[m3].size - 1)] [m, ( (m = m3 ⊃ (cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= (cache_dirs[m3].size - 1);cache_dirs[m]) .entries:= (m = m3 ⊃ (cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= (cache_dirs[m3].size - 1);cache_dirs[m]).entries [str0, entries1[str] ]) .size:= ((m = m3 ⊃ (cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= (cache_dirs[m3].size - 1);cache_dirs[m]).size + 1) ], cache_dirs[m3].size ≠ 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), str ∈ cache_dirs[m3].entries, m3 ∈ cache_dirs, entries1[str] ∈ cache_files, m ∈ cache_dirs, str ∈ entries1, 1 ≤ cache_dirs[m3].size, ¬ str0 ∈ entries0, ¬ entries1[str] ∈ cache_dirs, ¬ str0 ∈ cache_dirs[m].entries, 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, cache_dirs[m3].size); 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 := ?] ( afs-cons(dirs, cache_files) → ⟪new_dent := mkdentry(str0, mkdentry(str, entries1[str]).ino); old_dent1 := negdentry(mkdentry(str, entries1[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)}⟫ ⟨old_dent0 := mkdentry(str, entries1[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, cache_dirs[m3].size); old_child_inode0 := mkinode(entries1[str], md1, false, # links(entries1[str], cache_dirs), 0, m10); cache_dirs0 := 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(entries1[str], md1, false, # links(entries1[str], cache_dirs0), 0, m10) = 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

Rule argument

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

Additional information

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