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Proof step 702 from proof refine-rename-dir-dentry (previous step)

# links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(entries1[str0], cache_dirs) = cache_dirs[entries1[str0]].nlink, cache_dirs[entries0[str]].entries = ∅, cache_dirs[m3].size = m0 + 1, cache_dirs[m3] = mkdir(m0 + 1, md7, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md4, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), cache_dirs[entries1[str0]] = mkdir(cache_dirs[entries1[str0]].size, md3, cache_dirs[entries1[str0]].nlink, cache_dirs[entries1[str0]].nsubdirs, entries2), entries0[str] = cache_dirs[m].entries[str], entries1[str0] = cache_dirs[m3].entries[str0], afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= (cache_dirs[m3].nsubdirs - 1)] [m, (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;cache_dirs[m] ) .entries:= (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= (cache_dirs[m3].nsubdirs - 1) ;cache_dirs[m] ) .entries [str, entries1[str0] ] ], cache_dirs[m3].nsubdirs ≠ 0, entries0[str] ≠ ROOT_INO, 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), entries1[str0] ∈ cache_dirs, str0 ∈ entries1, m3 ∈ cache_dirs, m ∈ cache_dirs, entries0[str] ∈ cache_dirs, str ∈ cache_dirs[m].entries, str0 ∈ cache_dirs[m3].entries, str ∈ entries0, 1 ≤ cache_dirs[m3].nsubdirs, ⌊err0⌋ , m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ (err0 = err ∧ (asynced ↔ asynced2)), ⟨afs_drop_nlink#(mkdentry(str, entries0[str]).ino; dirs, afs_files); old_dent := mkdentry(str0, entries1[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries0[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, m0 + 1); dirs, afs_files); afs_inode_sync#(mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, afs_files); new_child_inode := mkinode(entries0[str], md1, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, afs_files); new_child_inode := (new_child_inode.ino ∈ dirs ⊃ mkinode (new_child_inode.ino, dirs[new_child_inode.ino].meta, true, dirs[new_child_inode.ino].nlink, dirs[new_child_inode.ino].nsubdirs, dirs[new_child_inode.ino].size) ;mkinode (new_child_inode.ino, afs_files[new_child_inode.ino].meta, false, afs_files[new_child_inode.ino].nlink, 0, afs_files[new_child_inode.ino].size) )}; old_parent_inode := (m3 ∈ dirs ⊃ mkinode(m3, dirs[m3].meta, true, dirs[m3].nlink, dirs[m3].nsubdirs, dirs[m3].size) ;mkinode(m3, afs_files[m3].meta, false, afs_files[m3].nlink, 0, afs_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, afs_files[m].meta, false, afs_files[m].nlink, 0, afs_files[m].size) ); err := ESUCCESS; asynced := ?⟩ ( mkinode(m3, md0, true, # links(m3, cache_dirs), cache_dirs[m3].nsubdirs, m0 + 1) = old_parent_inode ∧ mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size) = new_parent_inode ∧ mkinode (entries0[str], md1, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size) = new_child_inode ∧ mkdentry(str0, entries1[str0]) = old_dent ∧ mkdentry(str, entries0[str]) = new_dent ∧ err0 = err ∧ dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (asynced ↔ asynced2)), m3 = m ∧ mkdentry(str, entries0[str]) = mkdentry(str0, entries1[str0])

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 703:
# links(entries1[str0], cache_dirs) = cache_dirs[entries1[str0]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries0[str], cache_dirs) = cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].entries = ∅, cache_dirs[m3].nsubdirs = m1 + 1, cache_dirs[m3].size = m0 + 1, entries1[str0] = cache_dirs[m3].entries[str0], entries0[str] = cache_dirs[m].entries[str], cache_dirs[entries1[str0]] = mkdir(cache_dirs[entries1[str0]].size, md3, cache_dirs[entries1[str0]].nlink, cache_dirs[entries1[str0]].nsubdirs, entries2), cache_dirs[entries0[str]] = mkdir(cache_dirs[entries0[str]].size, md4, cache_dirs[entries0[str]].nlink, cache_dirs[entries0[str]].nsubdirs, ∅), cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_dirs[m3] = mkdir(m0 + 1, md7, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries1), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= m1] [m, (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= m1;cache_dirs[m]) .entries:= (m = m3 ⊃ ((cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str0)).size:= m0).nsubdirs:= m1;cache_dirs[m]).entries [str, entries1[str0] ] ], 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, str0 ∈ cache_dirs[m3].entries, str ∈ cache_dirs[m].entries, entries0[str] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, str0 ∈ entries1, entries1[str0] ∈ cache_dirs, 1 ≤ cache_dirs[m3].nsubdirs, ⌊err0⌋ , cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ (err0 = err ∧ (asynced ↔ asynced2)), ⟨afs_drop_nlink#(mkdentry(str, entries0[str]).ino; dirs, afs_files); old_dent := mkdentry(str0, entries1[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries0[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, m0 + 1); dirs, afs_files); afs_inode_sync#(mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size); dirs, afs_files); new_child_inode := mkinode(entries0[str], md1, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size); if asynced1 then {afs_inode_sync#(new_child_inode; dirs, afs_files); new_child_inode := (new_child_inode.ino ∈ dirs ⊃ mkinode (new_child_inode.ino, dirs[new_child_inode.ino].meta, true, dirs[new_child_inode.ino].nlink, dirs[new_child_inode.ino].nsubdirs, dirs[new_child_inode.ino].size) ;mkinode (new_child_inode.ino, afs_files[new_child_inode.ino].meta, false, afs_files[new_child_inode.ino].nlink, 0, afs_files[new_child_inode.ino].size) )}; old_parent_inode := (m3 ∈ dirs ⊃ mkinode(m3, dirs[m3].meta, true, dirs[m3].nlink, dirs[m3].nsubdirs, dirs[m3].size) ;mkinode(m3, afs_files[m3].meta, false, afs_files[m3].nlink, 0, afs_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, afs_files[m].meta, false, afs_files[m].nlink, 0, afs_files[m].size) ); err := ESUCCESS; asynced := ?⟩ ( mkinode(m3, md0, true, # links(m3, cache_dirs), cache_dirs[m3].nsubdirs, m0 + 1) = old_parent_inode ∧ mkinode(m, md, true, # links(m, cache_dirs), cache_dirs[m].nsubdirs, cache_dirs[m].size) = new_parent_inode ∧ mkinode (entries0[str], md1, true, # links(entries0[str], cache_dirs), cache_dirs[entries0[str]].nsubdirs, cache_dirs[entries0[str]].size) = new_child_inode ∧ mkdentry(str0, entries1[str0]) = old_dent ∧ mkdentry(str, entries0[str]) = new_dent ∧ err0 = err ∧ dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (asynced ↔ asynced2)), cache_dirs[m3].nsubdirs = 0, entries0[str] = ROOT_INO, m3 = m ∧ mkdentry(str, entries0[str]) = mkdentry(str0, entries1[str0])

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