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Proof step 127 from proof refine-unlink (previous step)

# 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_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries[str]).name)] [p_ino, cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries[str]).name)][p_ino] .size:= (cache_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries[str]).name)][p_ino].size - 1) ], ino = mkdentry(str, entries[str]).ino, p_ino = m3, mkdentry(str, entries[str]).dentry?, 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_dirs[p_ino, cache_dirs[p_ino].entries:= (cache_dirs[p_ino].entries -- mkdentry(str, entries[str]).name)][p_ino].size ≥ 1, 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 ⊦ ⟨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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 128:
# 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_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m3, (cache_dirs[m3].entries:= (cache_dirs[m3].entries -- str)).size:= m9], 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 ⊦ ⟨afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, afs_files); dent := negdentry(mkdentry(str, entries[str]).name); afs_inode_sync#(mkinode(m3, md1, true, cache_dirs[m3].nlink, 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: ⊦ m ≤ m + n; (s)% ⊦ m + 1 ≠ 0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n + 0 = n; (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat% ⊦ mkdentry(str, m).dentry?; (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)% ⊦ (ad.entries:= entries).size = ad.size; (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)% ⊦ mkdentry(str, m).name = str; (s)