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Proof step 58 from proof refine-unlink (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries0[str], cache_dirs) = cache_files[entries0[str]].nlink, entries0[str] = cache_dirs[m].entries[str], cache_files[entries0[str]] = mkfile(m0, md4, cache_files[entries0[str]].nlink, content1), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m7, md2, cache_files[entries0[str]].nlink, content0), cache_dirs[m] = mkdir(cache_dirs[m].size, md1, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, (cache_dirs[m].entries:= (cache_dirs[m].entries -- str)).size:= (cache_dirs[m].size - 1)], cache_dirs[m].size ≠ 0, 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[m].entries, entries0[str] ∈ cache_files, str ∈ entries0, m ∈ cache_dirs, 1 ≤ cache_dirs[m].size, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries0[str]).ino; dirs, cache_files); dent := negdentry(mkdentry(str, entries0[str]).name); afs_inode_sync#(mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, cache_files); afs_inode_sync#(mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); dirs, cache_files); p_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) ); c_inode := getinode(entries0[str], dirs, cache_files); err := ESUCCESS; asynced := ?; if err = ESUCCESS then let asynced = false in {icache_set#(c_inode, asynced; err, cache_icache); icache_set#(p_inode, asynced; err, cache_icache); dcache_delete#(p_inode.ino, dent.name; err, cache_dcache)}] ( icache-cons(cache_icache, dirs, cache_files) ∧ dcache-cons(cache_dcache, dirs, cache_files) ∧ pcache-cons(cache_pcache, dirs, cache_files) ∧ synced(dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) ∧ afs-cons(dirs, cache_files) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); p_inode1 := mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); afs_unlink#(; p_inode0, p_inode1, dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 59:
# links(entries0[str], cache_dirs) = cache_files[entries0[str]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, cache_dirs[m].size = m1 + 1, cache_dirs[m] = mkdir(cache_dirs[m].size, md1, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m7, md2, cache_files[entries0[str]].nlink, content0), cache_files[entries0[str]] = mkfile(m0, md4, cache_files[entries0[str]].nlink, content1), entries0[str] = cache_dirs[m].entries[str], afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, (cache_dirs[m].entries:= (cache_dirs[m].entries -- str)).size:= m1], 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), m ∈ cache_dirs, str ∈ entries0, entries0[str] ∈ cache_files, str ∈ cache_dirs[m].entries, 1 ≤ cache_dirs[m].size, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ [afs_drop_nlink#(mkdentry(str, entries0[str]).ino; dirs, cache_files); dent := negdentry(mkdentry(str, entries0[str]).name); afs_inode_sync#(mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, cache_files); afs_inode_sync#(mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); dirs, cache_files); p_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) ); c_inode := getinode(entries0[str], dirs, cache_files); err := ESUCCESS; asynced := ?; if err = ESUCCESS then let asynced = false in {icache_set#(c_inode, asynced; err, cache_icache); icache_set#(p_inode, asynced; err, cache_icache); dcache_delete#(p_inode.ino, dent.name; err, cache_dcache)}] ( icache-cons(cache_icache, dirs, cache_files) ∧ dcache-cons(cache_dcache, dirs, cache_files) ∧ pcache-cons(cache_pcache, dirs, cache_files) ∧ synced(dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) ∧ afs-cons(dirs, cache_files) → ⟨dent0 := mkdentry(str, entries0[str]); p_inode0 := mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); p_inode1 := mkinode(entries0[str], md0, false, # links(entries0[str], cache_dirs), 0, m6); afs_unlink#(; p_inode0, p_inode1, dent0, err0, cache_dirs, afs_files; afs_asynced)⟩ ( cache_dirs = dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced) ∧ p_inode = p_inode0 ∧ c_inode = p_inode1 ∧ dent = dent0 ∧ err = err0)), cache_dirs[m].size = 0

Rule argument

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

Additional information

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