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Proof step 155 from proof inv-mkdir (previous step)

dirs0[m0] = mkdir(m3, md1, # links(m0, dirs0), m5, entries0), afs-cons(dirs0, files), afs-cons ((dirs0[m0, mkdir(m3 + 1, md1, # links(m0, dirs0), m5 + 1, entries0[str, ino0])] ++ ino0)[ino0, mkdir(0, md, 1, 0, ∅)] [m0, mkdir(m3 + 1, md2, # links(m0, dirs0), m5 + 1, entries0[str, ino0]) ], files), dcache-cons(dcache, dirs0, files), synced(dirs0, icache0, pcache), m0 ∈ dirs0, asynced1 → synced(files, icache0, pcache), links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ ino0 = 0, ino0 ∈ dirs0, ino0 ∈ files, str ∈ entries0, ∀ key. key ∈ dcache[dentrykey(m0, str), mkdentry(str, ino0)] → key.dentry? ∧ key.name = dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].name ∧ valid-dentry (key.ino, dcache[dentrykey(m0, str), mkdentry(str, ino0)][key], (dirs0[m0, mkdir(m3 + 1, md1, # links(m0, dirs0), m5 + 1, entries0[str, ino0])] ++ ino0)[ino0, mkdir(0, md, 1, 0, ∅)] [m0, mkdir(m3 + 1, md2, # links(m0, dirs0), m5 + 1, entries0[str, ino0]) ], files)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 156:
dirs0[m0] = mkdir(m3, md1, # links(m0, dirs0), m5, entries0), afs-cons ((dirs0[m0, mkdir(m3 + 1, md1, # links(m0, dirs0), m5 + 1, entries0[str, ino0])] ++ ino0)[ino0, mkdir(0, md, 1, 0, ∅)] [m0, mkdir(m3 + 1, md2, # links(m0, dirs0), m5 + 1, entries0[str, ino0]) ], files), afs-cons(dirs0, files), dcache-cons(dcache, dirs0, files), synced(dirs0, icache0, pcache), key ∈ dcache[dentrykey(m0, str), mkdentry(str, ino0)], m0 ∈ dirs0, asynced1 → synced(files, icache0, pcache), links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ ino0 = 0, str ∈ entries0, ino0 ∈ files, ino0 ∈ dirs0, key.dentry? ∧ key.name = dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].name ∧ (key.ino = ino0 ∨ key.ino ∈ dirs0) ∧ ( links (key.ino, (dirs0[m0, mkdir(m3 + 1, md1, # links(m0, dirs0), m5 + 1, entries0[str, ino0])] ++ ino0)[ino0, mkdir(0, md, 1, 0, ∅)] [m0, mkdir(m3 + 1, md2, # links(m0, dirs0), m5 + 1, entries0[str, ino0]) ]) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ ( dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].ino = ino0 ∨ dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].ino ∈ dirs0 ∨ dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].ino ∈ files) ∧ dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].name ∈ (key.ino = m0 ⊃ mkdir(m3 + 1, md2, # links(key.ino, dirs0), m5 + 1, entries0[str, ino0]) ;(key.ino = ino0 ⊃ mkdir(0, md, 1, 0, ∅);(dirs0[m0, mkdir(m3 + 1, md1, # links(m0, dirs0), m5 + 1, entries0[str, ino0])] ++ ino0)[key.ino]) ) .entries ∧ dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].ino = (key.ino = m0 ⊃ mkdir(m3 + 1, md2, # links(key.ino, dirs0), m5 + 1, entries0[str, ino0]) ;(key.ino = ino0 ⊃ mkdir(0, md, 1, 0, ∅);(dirs0[m0, mkdir(m3 + 1, md1, # links(m0, dirs0), m5 + 1, entries0[str, ino0])] ++ ino0)[key.ino]) ) .entries [dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].name ] ∧ dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].dentry?

Rule argument

Additional information

Used simplifier rules: ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)%ino ≠ 0, ¬ ino ∈ dirs ⊦ ino0 ∈ dirs ++ ino ↔ ino0 = ino ∨ ino0 ∈ dirs; (s)% ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (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)