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Proof step 476 from proof refine-rename-dir-dentry (previous step)

(m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0) ) = mkdir(m, md1, m2, m5, entries), cache_dirs[m0] = mkdir(m1, md3, # links(m0, cache_dirs), m4, entries0), cache_dirs[entries0[str0]] = mkdir(m3, md4, # links(entries0[str0], cache_dirs), m6, ∅), cache_icache0 = cache_icache[m0, mkientry(false, mkinode(m0, md1, true, m2, m5, m))], err = ESUCCESS, entries0[str0] ≠ 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 ∈ entries0, m0 ∈ cache_dirs, entries0[str0] ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m0, cache_dirs) ≠ ∅ ∨ m0 = ROOT_INO ⊦ [icache_set# (mkinode (entries0[str0], cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)][entries0[str0]].meta:= md ] [entries0[str0] ] .meta, true, cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)][entries0[str0]].meta:= md ] [entries0[str0] ] .nlink, cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)][entries0[str0]].meta:= md ] [entries0[str0] ] .nsubdirs, cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)][entries0[str0]].meta:= md ] [entries0[str0] ] .size), false ; err, cache_icache0); dcache_delete#(m0, mkdentry(str0, entries0[str0]).name; err, cache_dcache); dcache_set# (m0, mkdentry(mkdentry(str0, entries0[str0]).name, mkdentry(str0, entries0[str0]).ino).name, mkdentry(mkdentry(str0, entries0[str0]).name, mkdentry(str0, entries0[str0]).ino) ; err, cache_dcache)] ( icache-cons (cache_icache0, cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_files) ∧ dcache-cons (cache_dcache, cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_files) ∧ pcache-cons (cache_pcache, cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_files) ∧ afs-cons (cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_files) ∧ synced (cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_icache0, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache0, cache_pcache)) → ⟨new_dent0 := mkdentry(str0, entries0[str0]); old_dent0 := mkdentry(str0, entries0[str0]); new_parent_inode0 := mkinode(m0, md0, true, # links(m0, cache_dirs), m4, m1); old_parent_inode0 := mkinode(m0, md0, true, # links(m0, cache_dirs), m4, m1); new_child_inode0 := mkinode (entries0[str0], md, true, # links(entries0[str0], cache_dirs), cache_dirs[entries0[str0]].nsubdirs, cache_dirs[entries0[str0]].size); old_child_inode0 := mkinode(entries0[str0], md2, true, # links(entries0[str0], cache_dirs), m6, m3); cache_dirs0 := cache_dirs; cache_dirs1 := cache_dirs; cache_dirs2 := cache_dirs; cache_dirs3 := cache_dirs; cache_dirs4 := cache_dirs; cache_dirs5 := cache_dirs; afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; cache_dirs3 := 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)⟩ ( (m0 ∈ cache_dirs5 ⊃ mkinode (m0, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .meta, true, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .nlink, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .nsubdirs, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .size) ;mkinode(m0, cache_files[m0].meta, false, cache_files[m0].nlink, 0, cache_files[m0].size) ) = old_parent_inode0 ∧ (m0 ∈ cache_dirs5 ⊃ mkinode (m0, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .meta, true, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .nlink, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .nsubdirs, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .size) ;mkinode(m0, cache_files[m0].meta, false, cache_files[m0].nlink, 0, cache_files[m0].size) ) = new_parent_inode0 ∧ mkinode (entries0[str0], md2, true, # links(entries0[str0], cache_dirs0), cache_dirs1[entries0[str0]].nsubdirs, cache_dirs2[entries0[str0]].size) = old_child_inode0 ∧ (entries0[str0] ∈ cache_dirs3 ⊃ mkinode (entries0[str0], md, true, (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs3), m4, entries0);cache_dirs3[entries0[str0]]).nlink, (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs3), m4, entries0);cache_dirs3[entries0[str0]]).nsubdirs, (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs3), m4, entries0);cache_dirs3[entries0[str0]]).size) ;mkinode (entries0[str0], cache_files[entries0[str0]].meta, false, cache_files[entries0[str0]].nlink, 0, cache_files[entries0[str0]].size) ) = new_child_inode0 ∧ negdentry(str0) = old_dent0 ∧ mkdentry(str0, entries0[str0]) = new_dent0 ∧ err = err0 ∧ cache_dirs = cache_dirs5[m0, mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ∧ entries0[str0] ∈ cache_dirs5 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs5), m4, entries0) ;cache_dirs5[entries0[str0]] ) .meta:= md ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache0 ∧ (afs_asynced ↔ asynced0)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 477:
(m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0) ) = mkdir(m, md1, m2, m5, entries), cache_dirs[entries0[str0]] = mkdir(m3, md4, # links(entries0[str0], cache_dirs), m6, ∅), cache_dirs[m0] = mkdir(m1, md3, # links(m0, cache_dirs), m4, entries0), cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)][entries0[str0]].meta:= md ] [entries0[str0] ] = mkdir(m7, md5, m8, m9, entries1), cache_icache0 = cache_icache[m0, mkientry(false, mkinode(m0, md1, true, m2, m5, m))], err = ESUCCESS, 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[str0] ∈ cache_dirs, m0 ∈ cache_dirs, str0 ∈ entries0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m0, cache_dirs) ≠ ∅ ∨ m0 = ROOT_INO ⊦ [icache_set#(mkinode(entries0[str0], md5, true, m8, m9, m7), false; err, cache_icache0); dcache_delete#(m0, mkdentry(str0, entries0[str0]).name; err, cache_dcache); dcache_set# (m0, mkdentry(mkdentry(str0, entries0[str0]).name, mkdentry(str0, entries0[str0]).ino).name, mkdentry(mkdentry(str0, entries0[str0]).name, mkdentry(str0, entries0[str0]).ino) ; err, cache_dcache)] ( icache-cons (cache_icache0, cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_files) ∧ dcache-cons (cache_dcache, cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_files) ∧ pcache-cons (cache_pcache, cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_files) ∧ afs-cons (cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_files) ∧ synced (cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs), m4, entries0);cache_dirs[entries0[str0]]).meta:= md ], cache_icache0, cache_pcache) ∧ (cache_sync → synced(cache_files, cache_icache0, cache_pcache)) → ⟨new_dent0 := mkdentry(str0, entries0[str0]); old_dent0 := mkdentry(str0, entries0[str0]); new_parent_inode0 := mkinode(m0, md0, true, # links(m0, cache_dirs), m4, m1); old_parent_inode0 := mkinode(m0, md0, true, # links(m0, cache_dirs), m4, m1); new_child_inode0 := mkinode (entries0[str0], md, true, # links(entries0[str0], cache_dirs), cache_dirs[entries0[str0]].nsubdirs, cache_dirs[entries0[str0]].size); old_child_inode0 := mkinode(entries0[str0], md2, true, # links(entries0[str0], cache_dirs), m6, m3); cache_dirs0 := cache_dirs; cache_dirs1 := cache_dirs; cache_dirs2 := cache_dirs; cache_dirs3 := cache_dirs; cache_dirs4 := cache_dirs; cache_dirs5 := cache_dirs; afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; cache_dirs3 := 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)⟩ ( (m0 ∈ cache_dirs5 ⊃ mkinode (m0, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .meta, true, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .nlink, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .nsubdirs, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .size) ;mkinode(m0, cache_files[m0].meta, false, cache_files[m0].nlink, 0, cache_files[m0].size) ) = old_parent_inode0 ∧ (m0 ∈ cache_dirs5 ⊃ mkinode (m0, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .meta, true, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .nlink, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .nsubdirs, (m0 = entries0[str0] ⊃ mkdir(m1, md, # links(entries0[str0], cache_dirs5), m4, entries0) ;mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0) ) .size) ;mkinode(m0, cache_files[m0].meta, false, cache_files[m0].nlink, 0, cache_files[m0].size) ) = new_parent_inode0 ∧ mkinode (entries0[str0], md2, true, # links(entries0[str0], cache_dirs0), cache_dirs1[entries0[str0]].nsubdirs, cache_dirs2[entries0[str0]].size) = old_child_inode0 ∧ (entries0[str0] ∈ cache_dirs3 ⊃ mkinode (entries0[str0], md, true, (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs3), m4, entries0);cache_dirs3[entries0[str0]]).nlink, (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs3), m4, entries0);cache_dirs3[entries0[str0]]).nsubdirs, (entries0[str0] = m0 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs3), m4, entries0);cache_dirs3[entries0[str0]]).size) ;mkinode (entries0[str0], cache_files[entries0[str0]].meta, false, cache_files[entries0[str0]].nlink, 0, cache_files[entries0[str0]].size) ) = new_child_inode0 ∧ negdentry(str0) = old_dent0 ∧ mkdentry(str0, entries0[str0]) = new_dent0 ∧ err = err0 ∧ cache_dirs = cache_dirs5[m0, mkdir(m1, md0, # links(m0, cache_dirs5), m4, entries0)] [entries0[str0], (entries0[str0] = m0 ∧ entries0[str0] ∈ cache_dirs5 ⊃ mkdir(m1, md0, # links(entries0[str0], cache_dirs5), m4, entries0) ;cache_dirs5[entries0[str0]] ) .meta:= md ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache0 ∧ (afs_asynced ↔ asynced0))), entries0[str0] = ROOT_INO

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] → [cache_dirs [m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0) ] [entries0[str0], cache_dirs [m0, mkdir (m1, md0, # links(m0, cache_dirs), m4, entries0) ] [entries0[str0] ] .meta:= md ] [entries0[str0] ], m7, md5, m8, m9, 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: cache_dirs[m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0)] [entries0[str0], cache_dirs [m0, mkdir(m1, md0, # links(m0, cache_dirs), m4, entries0) ] [entries0[str0] ] .meta:= md ] [entries0[str0] ], m7, md5, m8, m9, entries1