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Proof step 157 from proof inv-mkdir (previous step)

(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?

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 158:
⊦ dcache-cons(dcache, dirs, files) ↔ (∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ valid-dentry(key.ino, dcache[key], dirs, files))
Proof step 159:
(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[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), synced(dirs0, icache0, pcache), key ∈ dcache[dentrykey(m0, str), mkdentry(str, ino0)], m0 ∈ dirs0, asynced1 → synced(files, icache0, pcache), ∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ valid-dentry(key.ino, dcache[key], dirs0, files), 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 ∈ 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: cache-base name: dcache-cons seq: ⊦ dcache-cons(dcache, dirs, files) ↔ (∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ valid-dentry(key.ino, dcache[key], dirs, files)) subst: [dcache, dirs, files] → [dcache, dirs0, files] flags: <all paths>

Additional information

Used terms: dcache, dirs0, files