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Proof step 27 from proof refine-renam (previous step)

# links(m3, cache_dirs) = m20, # links(m, cache_dirs) = m5, cache_dirs[negdentry(str0).ino] = mkdir(m11, md8, m22, m23, entries2), cache_dirs[m] = mkdir(m4, md6, # links(m, cache_dirs), m6, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[negdentry(str0).ino] = mkfile(m17, md4, m18, content0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m15, md3, m16, content), cache_dirs[m3] = mkdir(m19, md5, # links(m3, cache_dirs), m21, entries), cache_dirs[entries0[str]] = mkdir(m0, md7, # links(entries0[str], cache_dirs), m2, entries1), afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, err = err0, m12 = # links(entries0[str], cache_dirs), new_child_inode = mkinode(m7, md1, boolvar, m8, m9, m10), new_dent = negdentry(str0), new_parent_inode = mkinode(m3, md0, true, # links(m3, cache_dirs), m21, m19), old_child_inode = mkinode(entries0[str], md2, true, # links(entries0[str], cache_dirs), m2, m0), old_dent = mkdentry(str, entries0[str]), old_parent_inode = mkinode(m, md, true, # links(m, cache_dirs), m6, m4), 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), entries0[str] ∈ cache_dirs, m ∈ cache_dirs, str ∈ entries0, m3 ∈ cache_dirs, ¬ str0 ∈ entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), m3 = m → md0 = md, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨old_dent0 := mkdentry(str, entries0[str]); p_inode := mkinode(m, md, true, m5, m6, m4); p_inode0 := mkinode(m3, md0, true, m20, m21, m19); p_inode2 := mkinode(m7, md1, boolvar, m8, m9, m10); p_inode1 := mkinode(entries0[str], md2, true, m12, m2, m0); old_dent1 := negdentry(str0); afs_rename#(; p_inode, p_inode0, p_inode1, p_inode2, old_dent0, old_dent1, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ old_parent_inode = p_inode ∧ new_parent_inode = p_inode0 ∧ old_child_inode = p_inode1 ∧ new_child_inode = p_inode2 ∧ old_dent = old_dent0 ∧ new_dent = old_dent1 ∧ err = err0)

User applied rule insert program lemma

And obtained the following new goals

Proof step 28:
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), afs-cons(afs_dirs, afs_files), synced(cache_dirs, cache_icache, cache_pcache), cache_sync → synced(cache_files, cache_icache, cache_pcache), valid-dentry(old_parent_inode.ino, old_dent, afs_dirs, afs_files), valid-negdentry(new_parent_inode.ino, new_dent, afs_dirs, afs_files), valid-dir-inode(old_parent_inode, afs_dirs, afs_files), valid-dir-inode(new_parent_inode, afs_dirs, afs_files), valid-dir-inode(old_child_inode, afs_dirs, afs_files), old_child_inode.ino = old_dent.ino, new_parent_inode.ino = old_parent_inode.ino → new_parent_inode.meta = old_parent_inode.meta, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, afs_asynced ↔ cache_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 ⊦ ⟪cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨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 ↔ cache_asynced))
Proof step 29:
# links(m, cache_dirs) = m5, # links(m3, cache_dirs) = m20, old_child_inode.ino = old_dent.ino, cache_asynced ↔ cache_asynced, cache_sync0 ↔ cache_sync, afs_asynced0 ↔ cache_asynced, cache_dirs[entries0[str]] = mkdir(m0, md7, # links(entries0[str], cache_dirs), m2, entries1), cache_dirs[m3] = mkdir(m19, md5, # links(m3, cache_dirs), m21, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str]] = mkfile(m15, md3, m16, content), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[negdentry(str0).ino] = mkfile(m17, md4, m18, content0), cache_dirs[m] = mkdir(m4, md6, # links(m, cache_dirs), m6, entries0), cache_dirs[negdentry(str0).ino] = mkdir(m11, md8, m22, m23, entries2), afs_dirs = cache_dirs, afs_dirs0 = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, afs_files0 = afs_files, cache_dirs = cache_dirs, cache_pcache0 = cache_pcache, err = err0, err = err, err1 = err, m12 = # links(entries0[str], cache_dirs), new_child_inode = mkinode(m7, md1, boolvar, m8, m9, m10), new_child_inode = new_child_inode, new_child_inode1 = new_child_inode, new_dent = negdentry(str0), new_dent = new_dent, new_dent1 = new_dent, new_parent_inode = mkinode(m3, md0, true, # links(m3, cache_dirs), m21, m19), new_parent_inode = new_parent_inode, new_parent_inode1 = new_parent_inode, old_child_inode = mkinode(entries0[str], md2, true, # links(entries0[str], cache_dirs), m2, m0), old_child_inode = old_child_inode, old_child_inode1 = old_child_inode, old_dent = mkdentry(str, entries0[str]), old_dent = old_dent, old_dent2 = old_dent, old_parent_inode = mkinode(m, md, true, # links(m, cache_dirs), m6, m4), old_parent_inode = old_parent_inode, old_parent_inode1 = old_parent_inode, afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, afs_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), valid-dentry(old_parent_inode.ino, old_dent, cache_dirs, afs_files), valid-dir-inode(old_parent_inode, cache_dirs, afs_files), valid-dir-inode(new_parent_inode, cache_dirs, afs_files), valid-dir-inode(old_child_inode, cache_dirs, afs_files), valid-negdentry(new_parent_inode.ino, new_dent, cache_dirs, afs_files), m3 ∈ cache_dirs, str ∈ entries0, m ∈ cache_dirs, entries0[str] ∈ cache_dirs, m3 = m → md0 = md, cache_sync → synced(cache_files, cache_icache, cache_pcache), new_parent_inode.ino = old_parent_inode.ino → new_parent_inode.meta = old_parent_inode.meta, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ [cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)] ( ⟨afs_rename# (; old_parent_inode1, new_parent_inode1, old_child_inode1, new_child_inode1, old_dent2, new_dent1, err1, afs_dirs0, afs_files0; afs_asynced)⟩ ( old_parent_inode = old_parent_inode1 ∧ new_parent_inode = new_parent_inode1 ∧ old_child_inode = old_child_inode1 ∧ new_child_inode = new_child_inode1 ∧ old_dent = old_dent2 ∧ new_dent = new_dent1 ∧ err = err1 ∧ afs_dirs0 = cache_dirs ∧ afs_files0 = (cache_files ⊕ cache_pcache0) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced)) → ⟨old_dent0 := mkdentry(str, entries0[str]); p_inode := mkinode(m, md, true, m5, m6, m4); p_inode0 := mkinode(m3, md0, true, m20, m21, m19); p_inode2 := mkinode(m7, md1, boolvar, m8, m9, m10); p_inode1 := mkinode(entries0[str], md2, true, m12, m2, m0); old_dent1 := negdentry(str0); afs_rename#(; p_inode, p_inode0, p_inode1, p_inode2, old_dent0, old_dent1, err0, afs_dirs, afs_files; afs_asynced)⟩ ( afs_dirs = cache_dirs ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache ∧ (afs_asynced ↔ cache_asynced) ∧ old_parent_inode = p_inode ∧ new_parent_inode = p_inode0 ∧ old_child_inode = p_inode1 ∧ new_child_inode = p_inode2 ∧ old_dent = old_dent0 ∧ new_dent = old_dent1 ∧ err = err0)), str0 ∈ entries

Rule argument

spec: <toplevel> name: refine-rename-dir-negdentry seq: 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), afs-cons(afs_dirs, afs_files), synced(cache_dirs, cache_icache, cache_pcache), cache_sync → synced(cache_files, cache_icache, cache_pcache), valid-dentry(old_parent_inode.ino, old_dent, afs_dirs, afs_files), valid-negdentry (new_parent_inode.ino, new_dent, afs_dirs, afs_files), valid-dir-inode(old_parent_inode, afs_dirs, afs_files), valid-dir-inode(new_parent_inode, afs_dirs, afs_files), valid-dir-inode(old_child_inode, afs_dirs, afs_files), old_child_inode.ino = old_dent.ino, new_parent_inode.ino = old_parent_inode.ino → new_parent_inode.meta = old_parent_inode.meta, afs_dirs = cache_dirs, afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, afs_asynced ↔ cache_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 ⊦ ⟪cache_rename# (; old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err, cache_dirs, cache_files, cache_icache, cache_dcache, cache_asynced)⟫ ⟨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 ↔ cache_asynced)) subst: [cache_asynced, cache_dcache, cache_icache, cache_files, cache_dirs, err, new_dent, old_dent, new_child_inode, old_child_inode, new_parent_inode, old_parent_inode, cache_pcache, afs_files, cache_sync, afs_dirs, afs_asynced, old_parent_inode0, new_parent_inode0, old_child_inode0, new_child_inode0, old_dent0, new_dent0, err0] → [cache_asynced, cache_dcache, cache_icache, cache_files, cache_dirs, err, new_dent, old_dent, new_child_inode, old_child_inode, new_parent_inode, old_parent_inode, cache_pcache, afs_files, cache_sync, cache_dirs, cache_asynced, old_parent_inode, new_parent_inode, old_child_inode, new_child_inode, old_dent, new_dent, err] flags: List([[1,0,0]])

Additional information

Used simplifier rules: ⊦ mkinode(m, md, boolvar, m0, m1, m2).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)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (s)% ⊦ mkdentry(str, m).ino = m; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ mkdentry(str, m).name = str; (s)% ⊦ mkdentry(str, m).dentry?; (s)%⊦ valid-negdentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ ¬ dent.name ∈ dirs[ino].entries ∧ dent.negdentry?; (s)% ⊦ negdentry(str).name = str; (s)% ⊦ negdentry(str).negdentry?; (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-dir-ino(ino, dirs, files) ↔ ino ∈ dirs; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nlink = m0; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).size = m2; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nsubdirs = m1; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).directory ↔ boolvar; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).meta = md; (s)% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)% ⊦ mkdir(m, md, m0, m1, entries).nlink = m0; (s)% ⊦ mkdir(m, md, m0, m1, entries).size = m; (s)% ⊦ mkdir(m, md, m0, m1, entries).nsubdirs = m1; (s)