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Proof step 70 from proof refine-rename-file-negdentry (previous step)

# links(m, cache_dirs) = cache_dirs[m].nlink, cache_files[entries[str]] = mkfile(m2, md3, # links(entries[str], cache_dirs), content0), cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m9, md2, # links(entries[str], cache_dirs), content), cache_dirs[m3] = mkdir(m1 + 1, md4, # links(m3, cache_dirs), m5, entries), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, mkdir(m1, md4, # links(m3, cache_dirs), m5, entries -- str)] [m, ( (m = m3 ⊃ mkdir(m1, md4, # links(m3, cache_dirs), m5, entries -- str);cache_dirs[m]) .entries:= (m = m3 ⊃ mkdir(m1, md4, # links(m3, cache_dirs), m5, entries -- str);cache_dirs[m]).entries[str0, entries[str]]) .size:= ((m = m3 ⊃ mkdir(m1, md4, # links(m3, cache_dirs), m5, entries -- str);cache_dirs[m]).size + 1) ], 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), m ∈ cache_dirs, entries[str] ∈ cache_files, m3 ∈ cache_dirs, str ∈ entries, ¬ str0 ∈ entries0, ¬ entries[str] ∈ cache_dirs, ¬ str0 ∈ cache_dirs[m].entries, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ [afs_inode_sync#(mkinode(m3, md0, true, # links(m3, cache_dirs), m5, m1 + 1); 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, entries[str]).ino); old_dent1 := negdentry(mkdentry(str, entries[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, entries[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), m5, m1 + 1); old_child_inode0 := mkinode(entries[str], md1, false, # links(entries[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(entries[str], md1, false, # links(entries[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

Proof step 71:
# links(m, cache_dirs) = cache_dirs[m].nlink, (m = m3 ⊃ mkdir(m1, md4, # links(m3, cache_dirs), m5, entries -- str);cache_dirs[m]) = mkdir(m0, md6, m4, m6, entries1), cache_dirs[m3] = mkdir(m1 + 1, md4, # links(m3, cache_dirs), m5, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m9, md2, # links(entries[str], cache_dirs), content), cache_dirs[m] = mkdir(cache_dirs[m].size, md5, cache_dirs[m].nlink, cache_dirs[m].nsubdirs, entries0), cache_files[entries[str]] = mkfile(m2, md3, # links(entries[str], cache_dirs), content0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, mkdir(m1, md4, # links(m3, cache_dirs), m5, entries -- str)] [m, ((m = m3 ⊃ mkdir(m1, md4, # links(m3, cache_dirs), m5, entries -- str);cache_dirs[m]).entries:= entries1[str0, entries[str]]).size:= (m0 + 1) ], 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 ∈ entries, m3 ∈ cache_dirs, entries[str] ∈ cache_files, m ∈ cache_dirs, 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, # links(m3, cache_dirs), m5, m1 + 1); 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, entries[str]).ino); old_dent1 := negdentry(mkdentry(str, entries[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, entries[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), m5, m1 + 1); old_child_inode0 := mkinode(entries[str], md1, false, # links(entries[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(entries[str], md1, false, # links(entries[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))), str0 ∈ entries0, entries[str] ∈ cache_dirs, str0 ∈ cache_dirs[m].entries

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [(m = m3 ⊃ mkdir (m1, md4, # links(m3, cache_dirs), m5, entries -- str) ;cache_dirs[m] ), m0, md6, m4, m6, entries1]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: (m = m3 ⊃ mkdir(m1, md4, # links(m3, cache_dirs), m5, entries -- str) ;cache_dirs[m] ), m0, md6, m4, m6, entries1