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Proof step 86 from proof refine-mkdir (previous step)

# 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), cache_dirs0 = cache_dirs, dent = negdentry(str), m0 = cache_dirs[m1].size, m2 = cache_dirs[m1].nsubdirs, 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 ∈ entries, ¬ m5 ∈ cache_files, ¬ m5 ∈ cache_dirs, ¬ str ∈ cache_dirs[m1].entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m1, cache_dirs) ≠ ∅ ∨ m1 = ROOT_INO ⊦ ∃ ino. ¬ ino ∈ cache_dirs ∧ ¬ ino ∈ cache_files ∧ ino ≠ 0 ∧ ⟨afs_mkdir_ino#(m1, md3, ino; dent, err, cache_dirs); afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; 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)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 87:
# links(m1, cache_dirs) = m7, (cache_dirs[m1, mkdir(m6 + 1, md1, m7, m8 + 1, entries[str, m5])] ++ m5)[m1] = mkdir(m, md0, m3, m4, entries0), cache_dirs[m1] = mkdir(m6, md1, m7, m8, entries), cache_dirs[m1] = mkdir(m6, md, m7, m8, entries1), cache_dirs0 = cache_dirs, dent = negdentry(str), m0 = m6, m2 = m8, p_inode = mkinode(m1, md2, true, m7, m8, m6), 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), m1 ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m1, cache_dirs) ≠ ∅ ∨ m1 = ROOT_INO ⊦ ∃ ino. ¬ ino ∈ cache_dirs ∧ ¬ ino ∈ cache_files ∧ ino ≠ 0 ∧ ⟨afs_mkdir_ino#(m1, md3, ino; dent, err, cache_dirs); afs_files := (cache_files ⊕ cache_pcache) ⊕ cache_icache; 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), m5 = 0, str ∈ entries, m5 ∈ cache_files, m5 ∈ cache_dirs, str ∈ entries1

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[m1], m6, md, m7, m8, 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[m1], m6, md, m7, m8, entries1