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Proof step 83 from proof refine-mkdir (previous step)

cache_dirs[m1] = mkdir(m0, md1, # links(m1, cache_dirs), m2, entries), (cache_dirs[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs), m2 + 1, entries[str, m5])] ++ m5)[m1] = mkdir(m, md0, m3, m4, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, c_inode = p_inode1, cache_dirs0 = cache_dirs, dent = dent0, dent0 = negdentry(str), dirs = cache_dirs, err = err0, files = afs_files, md = md3, p_inode = p_inode0, p_inode0 = mkinode(m1, md2, true, # links(m1, cache_dirs), m2, m0), 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), pre-mkdir(p_inode, dent, md, dirs, files), synced(cache_dirs, cache_icache, cache_pcache), m1 ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m1, cache_dirs) ≠ ∅ ∨ m1 = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ ( dirs = (cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m5, mkdir(0, md3, 1, 0, ∅)] [m1, (m1 = m5 ∧ m5 ∈ cache_dirs0[m5, mkdir(m0 + 1, md1, # links(m5, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .meta:= md2 ] ∧ files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[m1, mkientry(false, mkinode(m1, md2, true, m3, m4, m))] [m5, mkientry (false, (m5 ∈ cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkinode (m5, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).meta, true, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).nlink, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).nsubdirs, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).size) ;mkinode(m5, cache_files[m5].meta, false, cache_files[m5].nlink, 0, cache_files[m5].size) )) ] ∧ (asynced ↔ asynced0) ∧ (m1 ∈ cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkinode (m1, md2, true, (m1 = m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .nlink, (m1 = m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .nsubdirs, (m1 = m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .size) ;mkinode(m1, cache_files[m1].meta, false, cache_files[m1].nlink, 0, cache_files[m1].size) ) = p_inode ∧ (m5 ∈ cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkinode (m5, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).meta, true, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).nlink, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).nsubdirs, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).size) ;mkinode(m5, cache_files[m5].meta, false, cache_files[m5].nlink, 0, cache_files[m5].size) ) = c_inode ∧ mkdentry(str, m5) = dent ∧ ESUCCESS = err), ⟨choose ino with ¬ ino ∈ dirs ∧ ¬ ino ∈ files ∧ ino ≠ 0 in afs_mkdir_ino#(p_inode.ino, md, ino; dent, err, dirs); afs_inode_sync#(p_inode; dirs, files); p_inode := getinode(p_inode.ino, dirs, files); c_inode := getinode(dent.ino, dirs, files); asynced := ?⟩ ( dirs = (cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m5, mkdir(0, md3, 1, 0, ∅)] [m1, (m1 = m5 ∧ m5 ∈ cache_dirs0[m5, mkdir(m0 + 1, md1, # links(m5, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .meta:= md2 ] ∧ files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[m1, mkientry(false, mkinode(m1, md2, true, m3, m4, m))] [m5, mkientry (false, (m5 ∈ cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkinode (m5, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).meta, true, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).nlink, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).nsubdirs, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).size) ;mkinode(m5, cache_files[m5].meta, false, cache_files[m5].nlink, 0, cache_files[m5].size) )) ] ∧ (asynced ↔ asynced0) ∧ (m1 ∈ cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkinode (m1, md2, true, (m1 = m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .nlink, (m1 = m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .nsubdirs, (m1 = m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .size) ;mkinode(m1, cache_files[m1].meta, false, cache_files[m1].nlink, 0, cache_files[m1].size) ) = p_inode ∧ (m5 ∈ cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkinode (m5, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).meta, true, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).nlink, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).nsubdirs, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).size) ;mkinode(m5, cache_files[m5].meta, false, cache_files[m5].nlink, 0, cache_files[m5].size) ) = c_inode ∧ mkdentry(str, m5) = dent ∧ ESUCCESS = err), m5 = 0, m5 ∈ cache_dirs, m5 ∈ cache_files, str ∈ entries

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 84:
# links(m1, cache_dirs) = cache_dirs[m1].nlink, cache_dirs[m1] = mkdir(cache_dirs[m1].size, md1, cache_dirs[m1].nlink, cache_dirs[m1].nsubdirs, entries), (cache_dirs[m1, mkdir(cache_dirs[m1].size + 1, md1, cache_dirs[m1].nlink, cache_dirs[m1].nsubdirs + 1, entries[str, m5])] ++ m5)[m1] = mkdir(m, md0, m3, m4, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, cache_dirs0 = cache_dirs, dent = negdentry(str), m0 = cache_dirs[m1].size, m2 = cache_dirs[m1].nsubdirs, md = md3, p_inode = mkinode(m1, md2, true, cache_dirs[m1].nlink, cache_dirs[m1].nsubdirs, cache_dirs[m1].size), m5 ≠ 0, 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), m1 ∈ cache_dirs, ¬ str ∈ cache_dirs[m1].entries, ¬ m5 ∈ cache_dirs, ¬ m5 ∈ cache_files, ¬ str ∈ entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m1, cache_dirs) ≠ ∅ ∨ m1 = ROOT_INO ⊦ ⟨choose ino with ¬ ino ∈ cache_dirs ∧ ¬ ino ∈ afs_files ∧ ino ≠ 0 in afs_mkdir_ino#(p_inode.ino, md, ino; dent, err, cache_dirs); afs_inode_sync#(p_inode; cache_dirs, afs_files); p_inode := (p_inode.ino ∈ cache_dirs ⊃ mkinode (p_inode.ino, cache_dirs[p_inode.ino].meta, true, cache_dirs[p_inode.ino].nlink, cache_dirs[p_inode.ino].nsubdirs, cache_dirs[p_inode.ino].size) ;mkinode(p_inode.ino, afs_files[p_inode.ino].meta, false, afs_files[p_inode.ino].nlink, 0, afs_files[p_inode.ino].size) ); c_inode := getinode(dent.ino, cache_dirs, afs_files); asynced := ?⟩ ( cache_dirs = (cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m5, mkdir(0, md3, 1, 0, ∅)] [m1, (m1 = m5 ∧ m5 ∈ cache_dirs0[m5, mkdir(m0 + 1, md1, # links(m5, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .meta:= md2 ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[m1, mkientry(false, mkinode(m1, md2, true, m3, m4, m))] [m5, mkientry (false, (m5 ∈ cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkinode (m5, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).meta, true, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).nlink, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).nsubdirs, (m5 = m1 ⊃ mkdir(m, md2, m3, m4, entries0);mkdir(0, md3, 1, 0, ∅)).size) ;mkinode(m5, cache_files[m5].meta, false, cache_files[m5].nlink, 0, cache_files[m5].size) )) ] ∧ (asynced ↔ asynced0) ∧ (m1 ∈ cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkinode (m1, md2, true, (m1 = m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .nlink, (m1 = m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .nsubdirs, (m1 = m5 ⊃ mkdir(0, md3, 1, 0, ∅) ;(cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5)[m1] ) .size) ;mkinode(m1, cache_files[m1].meta, false, cache_files[m1].nlink, 0, cache_files[m1].size) ) = p_inode ∧ (m5 ∈ cache_dirs0[m1, mkdir(m0 + 1, md1, # links(m1, cache_dirs0), m2 + 1, entries[str, m5])] ++ m5 ⊃ mkinode (m5, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).meta, true, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).nlink, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).nsubdirs, (m5 = m1 ⊃ mkdir(0, md2, 1, 0, ∅);mkdir(0, md3, 1, 0, ∅)).size) ;mkinode(m5, cache_files[m5].meta, false, cache_files[m5].nlink, 0, cache_files[m5].size) ) = c_inode ∧ mkdentry(str, m5) = dent ∧ ESUCCESS = err)

Rule argument

Additional information

Used simplifier rules: 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)% ⊦ mkdir(m, md, m0, m1, entries).meta = md; (s)%ino ≠ 0, ¬ ino ∈ dirs ⊦ ino0 ∈ dirs ++ ino ↔ ino0 = ino ∨ ino0 ∈ dirs; (s)% ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (s)% ⊦ mkdir(m, md, m0, m1, entries).nsubdirs = m1; (s)% ⊦ mkdir(m, md, m0, m1, entries).nlink = m0; (s)% ⊦ mkdir(m, md, m0, m1, entries).size = m; (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).size = m2; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).nlink = m0; (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)% ⊦ negdentry(str).negdentry?; (s)% ⊦ negdentry(str).name = str; (s)%⊦ valid-parent-ino(ino, dirs, files) ↔ ino ∈ dirs ∧ (links(ino, dirs) ≠ ∅ ∨ ino = ROOT_INO); (s)%⊦ valid-negdentry(ino, dent, dirs, files) ↔ valid-parent-ino(ino, dirs, files) ∧ ¬ dent.name ∈ dirs[ino].entries ∧ dent.negdentry?; (s)% ⊦ mkinode(m, md, boolvar, m0, m1, m2).ino = m; (s)%⊦ pre-mkdir(p_inode, dent, md, dirs, files) ↔ valid-negdentry(p_inode.ino, dent, dirs, files) ∧ valid-dir-inode(p_inode, dirs, files); (s)