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Proof step 120 from proof refine-unlink (previous step)

# links(entries[str], cache_dirs) = m0 + 1, cache_dirs[m3] = mkdir(m9 + 1, md4, # links(m3, cache_dirs), m8, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m7, md2, m0 + 1, content0), cache_files[entries[str]] = mkfile(m2, md3, m0 + 1, content), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = p_inode1, cache_dirs0 = cache_dirs, dent = dent0, dent0 = mkdentry(str, entries[str]), dirs = cache_dirs, err = err0, files = afs_files, p_inode = p_inode0, p_inode0 = mkinode(m3, md1, true, # links(m3, cache_dirs), m8, m9 + 1), p_inode1 = mkinode(entries[str], md0, false, m0 + 1, 0, m6), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m8, entries -- str)], cache_files[entries[str], mkfile(m6, md0, m0, content)]), dcache-cons(cache_dcache, cache_dirs, cache_files), dcache-cons (cache_dcache -- dentrykey(m3, str), cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m8, entries -- str)], cache_files[entries[str], mkfile(m6, md0, m0, content)]), icache-cons(cache_icache, cache_dirs, cache_files), icache-cons (cache_icache[entries[str], mkientry(false, mkinode(entries[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, # links(m3, cache_dirs), m8, m9)) ], cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m8, entries -- str)], cache_files[entries[str], mkfile(m6, md0, m0, content)]), pcache-cons(cache_pcache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m8, entries -- str)], cache_files[entries[str], mkfile(m6, md0, m0, content)]), pre-unlink(p_inode, c_inode, dent, dirs, files), synced(cache_dirs, cache_icache, cache_pcache), synced (cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), m8, entries -- str)], cache_icache[entries[str], mkientry(false, mkinode(entries[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, # links(m3, cache_dirs), m8, m9)) ], cache_pcache), m3 ∈ cache_dirs, entries[str] ∈ cache_files, str ∈ entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), cache_sync → synced (cache_files[entries[str], mkfile(m6, md0, m0, content)], cache_icache[entries[str], mkientry(false, mkinode(entries[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, # links(m3, cache_dirs), m8, m9)) ], cache_pcache), links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ ( dirs = cache_dirs0[m3, mkdir(m9, md1, # links(m3, cache_dirs0), m8, entries -- str)] ∧ files = (cache_files[entries[str], mkfile(m6, md0, m0, content)] ⊕ cache_pcache) ⊕ cache_icache[entries[str], mkientry(false, mkinode(entries[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, # links(m3, cache_dirs0), m8, m9)) ] ∧ (asynced ↔ asynced0) ∧ (m3 ∈ cache_dirs0 ⊃ mkinode(m3, md1, true, # links(m3, cache_dirs0), m8, m9) ;mkinode (m3, (m3 = entries[str] ⊃ mkfile(m6, md0, m0, content);cache_files[m3]).meta, false, (m3 = entries[str] ⊃ mkfile(m6, md0, m0, content);cache_files[m3]).nlink, 0, (m3 = entries[str] ⊃ mkfile(m6, md0, m0, content);cache_files[m3]).size) ) = p_inode ∧ (entries[str] ∈ cache_dirs0 ⊃ mkinode (entries[str], (entries[str] = m3 ⊃ mkdir(m9, md1, # links(entries[str], cache_dirs0), m8, entries -- str);cache_dirs0[entries[str]]).meta, true, (entries[str] = m3 ⊃ mkdir(m9, md1, # links(entries[str], cache_dirs0), m8, entries -- str);cache_dirs0[entries[str]]).nlink, (entries[str] = m3 ⊃ mkdir(m9, md1, # links(entries[str], cache_dirs0), m8, entries -- str);cache_dirs0[entries[str]]).nsubdirs, (entries[str] = m3 ⊃ mkdir(m9, md1, # links(entries[str], cache_dirs0), m8, entries -- str);cache_dirs0[entries[str]]).size) ;mkinode(entries[str], md0, false, m0, 0, m6) ) = c_inode ∧ negdentry(str) = dent ∧ ESUCCESS = err), ⟨let p_ino = p_inode.ino in {dirs[p_ino].entries := dirs[p_ino].entries -- dent.name; dirs[p_ino].size := dirs[p_ino].size - 1; let ino = dent.ino in afs_drop_nlink#(ino; dirs, files); dent := negdentry(dent.name); afs_inode_sync#(p_inode; dirs, files); afs_inode_sync#(c_inode; dirs, files); p_inode := getinode(p_inode.ino, dirs, files); c_inode := getinode(c_inode.ino, dirs, files); err := ESUCCESS}; asynced := ?⟩ ( dirs = cache_dirs0[m3, mkdir(m9, md1, # links(m3, cache_dirs0), m8, entries -- str)] ∧ files = (cache_files[entries[str], mkfile(m6, md0, m0, content)] ⊕ cache_pcache) ⊕ cache_icache[entries[str], mkientry(false, mkinode(entries[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, # links(m3, cache_dirs0), m8, m9)) ] ∧ (asynced ↔ asynced0) ∧ (m3 ∈ cache_dirs0 ⊃ mkinode(m3, md1, true, # links(m3, cache_dirs0), m8, m9) ;mkinode (m3, (m3 = entries[str] ⊃ mkfile(m6, md0, m0, content);cache_files[m3]).meta, false, (m3 = entries[str] ⊃ mkfile(m6, md0, m0, content);cache_files[m3]).nlink, 0, (m3 = entries[str] ⊃ mkfile(m6, md0, m0, content);cache_files[m3]).size) ) = p_inode ∧ (entries[str] ∈ cache_dirs0 ⊃ mkinode (entries[str], (entries[str] = m3 ⊃ mkdir(m9, md1, # links(entries[str], cache_dirs0), m8, entries -- str);cache_dirs0[entries[str]]).meta, true, (entries[str] = m3 ⊃ mkdir(m9, md1, # links(entries[str], cache_dirs0), m8, entries -- str);cache_dirs0[entries[str]]).nlink, (entries[str] = m3 ⊃ mkdir(m9, md1, # links(entries[str], cache_dirs0), m8, entries -- str);cache_dirs0[entries[str]]).nsubdirs, (entries[str] = m3 ⊃ mkdir(m9, md1, # links(entries[str], cache_dirs0), m8, entries -- str);cache_dirs0[entries[str]]).size) ;mkinode(entries[str], md0, false, m0, 0, m6) ) = c_inode ∧ negdentry(str) = dent ∧ ESUCCESS = err)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 121:
# links(m3, cache_dirs) = cache_dirs[m3].nlink, # links(entries[str], cache_dirs) = m0 + 1, m9 + 1 = cache_dirs[m3].size, entries[str] = cache_dirs[m3].entries[str], cache_dirs[m3] = mkdir(m9 + 1, md4, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m7, md2, m0 + 1, content0), cache_files[entries[str]] = mkfile(m2, md3, m0 + 1, content), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons (cache_dirs[m3, mkdir(m9, md1, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries -- str)], cache_files[entries[str], mkfile(m6, md0, m0, content)]), dcache-cons(cache_dcache, cache_dirs, cache_files), dcache-cons (cache_dcache -- dentrykey(m3, str), cache_dirs[m3, mkdir(m9, md1, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries -- str)], cache_files[entries[str], mkfile(m6, md0, m0, content)]), icache-cons(cache_icache, cache_dirs, cache_files), icache-cons (cache_icache[entries[str], mkientry(false, mkinode(entries[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, m9)) ], cache_dirs[m3, mkdir(m9, md1, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries -- str)], cache_files[entries[str], mkfile(m6, md0, m0, content)]), pcache-cons(cache_pcache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs[m3, mkdir(m9, md1, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries -- str)], cache_files[entries[str], mkfile(m6, md0, m0, content)]), synced(cache_dirs, cache_icache, cache_pcache), synced (cache_dirs[m3, mkdir(m9, md1, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, entries -- str)], cache_icache[entries[str], mkientry(false, mkinode(entries[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, m9)) ], cache_pcache), str ∈ cache_dirs[m3].entries, m3 ∈ cache_dirs, entries[str] ∈ cache_files, str ∈ entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), cache_sync → synced (cache_files[entries[str], mkfile(m6, md0, m0, content)], cache_icache[entries[str], mkientry(false, mkinode(entries[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, cache_dirs[m3].nlink, cache_dirs[m3].nsubdirs, m9)) ], cache_pcache), links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ ⟨let p_ino = m3 in {dirs := cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries[str]).name)]; dirs[p_ino].size := dirs[p_ino].size - 1; let ino = mkdentry(str, entries[str]).ino in {afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; afs_drop_nlink#(ino; dirs, afs_files)}; dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(m3, md1, true, # links(m3, cache_dirs), cache_dirs[m3].nsubdirs, m9 + 1); dirs, afs_files); afs_inode_sync#(mkinode(entries[str], md0, false, m0 + 1, 0, m6); dirs, afs_files); p_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) ); c_inode := getinode(entries[str], dirs, afs_files); err := ESUCCESS}; asynced := ?⟩ ( dirs = cache_dirs[m3, mkdir(m9, md1, # links(m3, cache_dirs), cache_dirs[m3].nsubdirs, entries -- str)] ∧ afs_files = (cache_files[entries[str], mkfile(m6, md0, m0, content)] ⊕ cache_pcache) ⊕ cache_icache[entries[str], mkientry(false, mkinode(entries[str], md0, false, m0, 0, m6))] [m3, mkientry(false, mkinode(m3, md1, true, # links(m3, cache_dirs), cache_dirs[m3].nsubdirs, m9)) ] ∧ (asynced ↔ asynced0) ∧ mkinode(m3, md1, true, # links(m3, cache_dirs), cache_dirs[m3].nsubdirs, m9) = p_inode ∧ mkinode(entries[str], md0, false, m0, 0, m6) = c_inode ∧ negdentry(str) = dent ∧ ESUCCESS = err)

Rule argument

Additional information

Used simplifier rules: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs → ¬ ino ∈ files; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)% ⊦ mkfile(m, md, m0, content).nlink = m0; (s)%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%⊦ getinode(ino, dirs, files) = (ino ∈ dirs ⊃ mkinode (ino, dirs[ino].meta, true, dirs[ino].nlink, dirs[ino].nsubdirs, dirs[ino].size) ;mkinode (ino, files[ino].meta, false, files[ino].nlink, 0, files[ino].size) ); (s)% ⊦ mkdentry(str, m) ≠ negdentry(str0); (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).size = m2; (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)% ⊦ 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).nlink = m0; (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)% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ mkdentry(str, m).name = str; (s)% ⊦ ino ∈ files ⊕ pcache ↔ ino ∈ files; (s)% ⊦ ino ∈ files ⊕ icache ↔ ino ∈ files; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ mkdentry(str, m).ino = m; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (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)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)%⊦ pre-unlink(p_inode, c_inode, dent, dirs, files) ↔ valid-dentry(p_inode.ino, dent, dirs, files) ∧ valid-file-inode(c_inode, dirs, files) ∧ valid-dir-inode(p_inode, dirs, files) ∧ c_inode.ino = dent.ino; (s)