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Proof step 37 from proof refine-rename-dir-dentry (previous step)

# links(m3, cache_dirs) = m12, # links(entries0[str0], cache_dirs) = m1, # links(m, cache_dirs) = m9, # links(entries2[str], cache_dirs) = m16, asynced0 ↔ new_dent.dentry?, asynced1 ↔ cache_asynced, cache_dirs[entries0[str0]] = mkdir(m8, md4, # links(entries0[str0], cache_dirs), m11, entries), cache_dirs[entries2[str]] = mkdir(m4, md7, # links(entries2[str], cache_dirs), m6, ∅), cache_dirs[m] = mkdir(m5, md3, # links(m, cache_dirs), m14, entries2), cache_dirs[m3] = mkdir(m0, md5, # links(m3, cache_dirs), m10, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dcache = cache_dcache, dirs = cache_dirs, err = err0, files = cache_files, icache = cache_icache, new_child_inode = mkinode(entries2[str], md1, true, # links(entries2[str], cache_dirs), m6, m4), new_dent = mkdentry(str, entries2[str]), new_parent_inode = mkinode(m, md, true, # links(m, cache_dirs), m14, m5), old_child_inode = mkinode(entries0[str0], md2, true, # links(entries0[str0], cache_dirs), m11, m8), old_dent = mkdentry(str0, entries0[str0]), old_parent_inode = mkinode(m3, md0, true, # links(m3, cache_dirs), m10, m0), 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), pre-rename(old_parent_inode, old_dent, old_child_inode, new_parent_inode, new_dent, new_child_inode, dirs, files), synced(cache_dirs, cache_icache, cache_pcache), str ∈ entries2, str0 ∈ entries0, entries2[str] ∈ cache_dirs, entries0[str0] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, 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 ⊦ [afs_rename#(; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, dirs, files; asynced1); if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, icache); icache_set#(new_parent_inode, asynced; err, icache); if asynced0 then icache_set#(new_child_inode, asynced; err, icache); dcache_delete#(old_parent_inode.ino, old_dent.name; err, dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, dcache)}] ( icache-cons(icache, dirs, files) ∧ dcache-cons(dcache, dirs, files) ∧ pcache-cons(cache_pcache, dirs, files) ∧ afs-cons(dirs, files) ∧ synced(dirs, icache, cache_pcache) ∧ (cache_sync → synced(files, icache, cache_pcache)) → ⟨new_dent0 := mkdentry(str, entries2[str]); old_dent0 := mkdentry(str0, entries0[str0]); new_parent_inode0 := mkinode(m, md, true, m9, m14, m5); old_parent_inode0 := mkinode(m3, md0, true, m12, m10, m0); new_child_inode0 := mkinode(entries2[str], md1, true, m16, m6, m4); old_child_inode0 := mkinode(entries0[str0], md2, true, m1, m11, m8); afs_rename# (; old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode0 ∧ new_parent_inode = new_parent_inode0 ∧ old_child_inode = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ afs_dirs = dirs ∧ afs_files = (files ⊕ cache_pcache) ⊕ icache ∧ (afs_asynced ↔ asynced1))), entries2[str] = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 38:
# links(entries0[str0], cache_dirs) = cache_dirs[entries0[str0]].nlink, # links(m, cache_dirs) = cache_dirs[m].nlink, # links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries2[str], cache_dirs) = cache_dirs[entries2[str]].nlink, cache_dirs[entries2[str]].entries = ∅, cache_dirs[m3].nlink = m12, cache_dirs[entries0[str0]].nlink = m1, cache_dirs[m].nlink = m9, cache_dirs[entries2[str]].nlink = m16, entries0[str0] = cache_dirs[m3].entries[str0], entries2[str] = cache_dirs[m].entries[str], cache_dirs[entries0[str0]] = mkdir(cache_dirs[entries0[str0]].size, md4, cache_dirs[entries0[str0]].nlink, cache_dirs[entries0[str0]].nsubdirs, entries), cache_dirs[entries2[str]] = mkdir(cache_dirs[entries2[str]].size, md7, cache_dirs[entries2[str]].nlink, cache_dirs[entries2[str]].nsubdirs, ∅), cache_dirs[m] = mkdir(cache_dirs[m].size, md3, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries2), cache_dirs[m3] = mkdir(cache_dirs[m3].size, md5, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries0), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m0 = cache_dirs[m3].size, m10 = cache_dirs[m3].nsubdirs, m11 = cache_dirs[entries0[str0]].nsubdirs, m14 = cache_dirs[m].nsubdirs, m4 = cache_dirs[entries2[str]].size, m5 = cache_dirs[m].size, m6 = cache_dirs[entries2[str]].nsubdirs, m8 = cache_dirs[entries0[str0]].size, new_child_inode = mkinode(entries2[str], md1, true, cache_dirs[entries2[str]].nlink, cache_dirs[entries2[str]].nsubdirs, cache_dirs[entries2[str]].size), new_dent = mkdentry(str, entries2[str]), new_parent_inode = mkinode(m, md, true, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, cache_dirs[m].size), old_child_inode = mkinode(entries0[str0], md2, true, cache_dirs[entries0[str0]].nlink, cache_dirs[entries0[str0]].nsubdirs, cache_dirs[entries0[str0]].size), old_dent = mkdentry(str0, entries0[str0]), old_parent_inode = mkinode(m3, md0, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, cache_dirs[m3].size), entries2[str] ≠ ROOT_INO, 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), str0 ∈ cache_dirs[m3].entries, str ∈ cache_dirs[m].entries, str ∈ entries2, str0 ∈ entries0, entries2[str] ∈ cache_dirs, entries0[str0] ∈ cache_dirs, m ∈ cache_dirs, m3 ∈ cache_dirs, 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 ⊦ [afs_rename#(; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files; asynced1); 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); if mkdentry(str, entries2[str]).dentry? then icache_set#(new_child_inode, asynced; err, cache_icache); dcache_delete#(old_parent_inode.ino, old_dent.name; err, cache_dcache); dcache_set#(new_parent_inode.ino, new_dent.name, new_dent; err, cache_dcache)}] ( icache-cons(cache_icache, cache_dirs, cache_files) ∧ dcache-cons(cache_dcache, cache_dirs, cache_files) ∧ pcache-cons(cache_pcache, cache_dirs, cache_files) ∧ afs-cons(cache_dirs, cache_files) ∧ synced(cache_dirs, cache_icache, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache, cache_pcache)) → ⟨new_dent0 := mkdentry(str, entries2[str]); old_dent0 := mkdentry(str0, entries0[str0]); new_parent_inode0 := mkinode(m, md, true, m9, m14, m5); old_parent_inode0 := mkinode(m3, md0, true, m12, m10, m0); new_child_inode0 := mkinode(entries2[str], md1, true, m16, m6, m4); old_child_inode0 := mkinode(entries0[str0], md2, true, m1, m11, m8); afs_rename# (; old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0, afs_dirs, afs_files; afs_asynced)⟩ ( old_parent_inode = old_parent_inode0 ∧ new_parent_inode = new_parent_inode0 ∧ old_child_inode = old_child_inode0 ∧ new_child_inode = new_child_inode0 ∧ old_dent = old_dent0 ∧ new_dent = new_dent0 ∧ err = err0 ∧ afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ asynced1)))

Rule argument

Additional information

Used simplifier rules: afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base% ⊦ mkinode(m, md, boolvar, m0, m1, m2).meta = md; (s)% ⊦ dir-empty(ino, dirs, files) ↔ dirs[ino].entries = ∅; (s)% ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs → ¬ ino ∈ files; (s)% ⊦ valid-file-ino(ino, dirs, files) ↔ ino ∈ files; (s)%⊦ valid-file-inode(inode, dirs, files) ↔ valid-file-ino(inode.ino, dirs, files) ∧ inode.nlink = files[inode.ino].nlink ∧ inode.nsubdirs = 0 ∧ ¬ inode.directory; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).directory ↔ boolvar; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nsubdirs = m1; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).size = m2; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nlink = m0; (s)% ⊦ valid-dir-ino(ino, dirs, files) ↔ ino ∈ dirs; (s)%⊦ valid-dir-inode(inode, dirs, files) ↔ valid-dir-ino(inode.ino, dirs, files) ∧ inode.nlink = dirs[inode.ino].nlink ∧ inode.size = dirs[inode.ino].size ∧ inode.nsubdirs = dirs[inode.ino].nsubdirs ∧ inode.directory; (s)%⊦ valid-wb-inode(inode, dirs, files) ↔ valid-dir-inode(inode, dirs, files) ∨ valid-file-inode(inode, dirs, files); (s)% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ mkdentry(str, m).ino = m; (s)%⊦ valid-dentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ valid-ino(dent.ino, dirs, files) ∧ dent.name ∈ dirs[ino].entries ∧ dent.ino = dirs[ino].entries[dent.name] ∧ dent.dentry?; (s)% ⊦ ¬ mkdentry(str, m).negdentry?; (s)% ⊦ mkdentry(str, m).name = str; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (s)%⊦ valid-negdentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ ¬ dent.name ∈ dirs[ino].entries ∧ dent.negdentry?; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)%⊦ pre-rename (old_parent_inode, old_dent, old_child_inode, new_parent_inode, new_dent, new_child_inode, dirs, files) ↔ ( valid-negdentry(new_parent_inode.ino, new_dent, dirs, files) ∨ valid-dentry(new_parent_inode.ino, new_dent, dirs, files) ∧ valid-wb-inode(new_child_inode, dirs, files) ∧ new_child_inode.ino = new_dent.ino ∧ ( valid-file-ino(new_dent.ino, dirs, files) ↔ valid-file-ino(old_dent.ino, dirs, files)) ∧ ( valid-dir-ino(new_dent.ino, dirs, files) → new_dent.ino ≠ ROOT_INO ∧ dir-empty(new_dent.ino, dirs, files))) ∧ valid-dentry(old_parent_inode.ino, old_dent, dirs, files) ∧ valid-dir-inode(old_parent_inode, dirs, files) ∧ valid-dir-inode(new_parent_inode, dirs, files) ∧ ( new_parent_inode.ino = old_parent_inode.ino → new_parent_inode.meta = old_parent_inode.meta) ∧ valid-wb-inode(old_child_inode, dirs, files) ∧ old_child_inode.ino = old_dent.ino; (s)