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Proof step 156 from proof inv-mkdir (previous step)

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?

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 157:
(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]) ) = mkdir(m, md0, m1, m2, entries), 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, key ∈ dcache[dentrykey(m0, str), mkdentry(str, ino0)], asynced1 → synced(files, icache0, pcache), links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ ino0 = 0, ino0 ∈ dirs0, ino0 ∈ files, str ∈ entries0, 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 ∈ entries ∧ dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].ino = entries[dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].name] ∧ dcache[dentrykey(m0, str), mkdentry(str, ino0)][key].dentry?

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] → [(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 ] ) ), m, md0, m1, m2, entries]

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: (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 ] ) ), m, md0, m1, m2, entries