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Proof step 153 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), m0 ∈ dirs0, asynced1 → synced(files, icache0, pcache), links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ ino0 = 0, dcache-cons (dcache[dentrykey(m0, str), mkdentry(str, ino0)], (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), str ∈ entries0, ino0 ∈ files, ino0 ∈ dirs0

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 154:
⊦ 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 155:
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)

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[dentrykey(m0, str), mkdentry(str, ino0)], ( 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] flags: <all paths>

Additional information

Used terms: dcache[dentrykey(m0, str), mkdentry(str, ino0)], ( 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