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Proof step 166 from proof inv-mkdir (previous step)

dcache[key].ino = entries1[key.name], key.name = dcache[key].name, (key.ino = m0 ⊃ mkdir(m3 + 1, md2, # links(key.ino, dirs0), m5 + 1, entries0[str, ino0]) ;mkdir(m4, md3, # links(key.ino, dirs0), m7, entries1) ) = mkdir(m, md0, m1, m2, entries), dirs0[key.ino] = mkdir(m4, md3, # links(key.ino, dirs0), m7, entries1), dirs0[m0] = mkdir(m3, md1, # links(m0, dirs0), m5, entries0), dcache[key].dentry?, key.dentry?, 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.name ∈ entries1, key.ino ∈ dirs0, key ∈ dcache, m0 ∈ dirs0, asynced1 → synced(files, icache0, pcache), ∀ key0. key0 ∈ dcache → key0.dentry? ∧ key0.name = dcache[key0].name ∧ key0.ino ∈ dirs0 ∧ (links(key0.ino, dirs0) ≠ ∅ ∨ key0.ino = ROOT_INO) ∧ (dcache[key0].ino ∈ dirs0 ∨ dcache[key0].ino ∈ files) ∧ dcache[key0].name ∈ dirs0[key0.ino].entries ∧ dcache[key0].ino = dirs0[key0.ino].entries[dcache[key0].name] ∧ dcache[key0].dentry?, entries1[key.name] ∈ dirs0 ∨ entries1[key.name] ∈ files, links(key.ino, dirs0) ≠ ∅ ∨ key.ino = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ ino0 = 0, key = dentrykey(m0, str), str ∈ entries0, ino0 ∈ files, ino0 ∈ 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) ∧ (entries1[key.name] = ino0 ∨ entries1[key.name] ∈ dirs0 ∨ entries1[key.name] ∈ files) ∧ key.name ∈ entries ∧ entries1[key.name] = entries[key.name]

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 167:
dcache[key].ino = entries1[str0], (m6 = m0 ⊃ mkdir(m3 + 1, md2, # links(m6, dirs0), m5 + 1, entries0[str, ino0]);mkdir(m4, md3, # links(m6, dirs0), m7, entries1)) = mkdir(m, md0, m1, m2, entries), dirs0[m0] = mkdir(m3, md1, # links(m0, dirs0), m5, entries0), dirs0[m6] = mkdir(m4, md3, # links(m6, dirs0), m7, entries1), key = dentrykey(m6, str0), str0 = dcache[key].name, key.dentry?, dcache[key].dentry?, 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), synced(dirs0, icache0, pcache), m0 ∈ dirs0, key ∈ dcache, m6 ∈ dirs0, str0 ∈ entries1, asynced1 → synced(files, icache0, pcache), ∀ key0. key0 ∈ dcache → key0.dentry? ∧ key0.name = dcache[key0].name ∧ key0.ino ∈ dirs0 ∧ (links(key0.ino, dirs0) ≠ ∅ ∨ key0.ino = ROOT_INO) ∧ (dcache[key0].ino ∈ dirs0 ∨ dcache[key0].ino ∈ files) ∧ dcache[key0].name ∈ dirs0[key0.ino].entries ∧ dcache[key0].ino = dirs0[key0.ino].entries[dcache[key0].name] ∧ dcache[key0].dentry?, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m6, dirs0) ≠ ∅ ∨ m6 = ROOT_INO, entries1[str0] ∈ dirs0 ∨ entries1[str0] ∈ files ⊦ ino0 = 0, key = dentrykey(m0, str), ino0 ∈ dirs0, ino0 ∈ files, str ∈ entries0, ( links (m6, (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]) ]) ≠ ∅ ∨ m6 = ROOT_INO) ∧ (entries1[str0] = ino0 ∨ entries1[str0] ∈ dirs0 ∨ entries1[str0] ∈ files) ∧ str0 ∈ entries ∧ entries1[str0] = entries[str0]

Rule argument

spec: key name: elim-03 seq: ⊦ key.dentry? → (m = key.ino ∧ str = key.name ↔ key = dentrykey(m, str)) subst: [key, m, str] → [key, m6, str0]

Additional information

Used simplifier rules: ⊦ key.dentry? → (m = key.ino ∧ str = key.name ↔ key = dentrykey(m, str)); (elim) Used terms: key, m6, str0