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Proof step 88 from proof inv-evict (previous step)

# links(m, dirs) = 0, dirs[m] = mkdir(m0, md, 0, m2, entries), dirs[key.ino] = mkdir(m1, md0, m3, m4, entries0), afs-cons(dirs, files), afs-cons(dirs -- m, files), dcache-cons(dcache, dirs, files), synced(dirs, icache, pcache), m ∈ dirs, key ∈ dcache, asynced1 → synced(files, icache, pcache) ⊦ m = ROOT_INO, key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ≠ m ∧ key.ino ∈ dirs ∧ (links(key.ino, dirs -- m) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ≠ m ∧ dcache[key].ino ∈ dirs ∨ dcache[key].ino ∈ files) ∧ dcache[key].name ∈ entries0 ∧ dcache[key].ino = entries0[dcache[key].name] ∧ dcache[key].dentry?

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 89:
⊦ 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 90:
# links(m, dirs) = 0, dirs[key.ino] = mkdir(m1, md0, m3, m4, entries0), dirs[m] = mkdir(m0, md, 0, m2, entries), afs-cons(dirs -- m, files), afs-cons(dirs, files), synced(dirs, icache, pcache), key ∈ dcache, m ∈ dirs, asynced1 → synced(files, icache, pcache), ∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ valid-dentry(key.ino, dcache[key], dirs, files) ⊦ m = ROOT_INO, key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ≠ m ∧ key.ino ∈ dirs ∧ (links(key.ino, dirs -- m) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ≠ m ∧ dcache[key].ino ∈ dirs ∨ dcache[key].ino ∈ files) ∧ dcache[key].name ∈ entries0 ∧ dcache[key].ino = entries0[dcache[key].name] ∧ dcache[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, dirs, files] flags: <all paths>

Additional information

Used terms: dcache, dirs, files