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Proof step 10 from proof inv-evict (previous step)

key ∈ evict(ino, pcache0[datakey(ino0, ino1), pe]), ∀ key. key ∈ pcache0[datakey(ino0, ino1), pe] → key.data?, ∀ key, ino. key ∈ evict(ino, pcache0) ∧ (∀ key. key ∈ pcache0 → key.data?) → key.data? ∧ key.ino ≠ ino ⊦ key.data? ∧ key.ino ≠ ino

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
n ≠ ino ⊦ evict(n, pcache[datakey(ino, pageno), pe]) = evict(n, pcache)[datakey(ino, pageno), pe]
Proof step 12:
ino = ino0, key ∈ evict(ino, pcache0[datakey(ino0, ino1), pe]), ∀ key, ino. key ∈ evict(ino, pcache0) ∧ (∀ key. key ∈ pcache0 → key.data?) → key.data? ∧ key.ino ≠ ino, ∀ key. key ∈ pcache0[datakey(ino0, ino1), pe] → key.data? ⊦ key.data? ∧ key.ino ≠ ino
Proof step 16:
key ∈ evict(ino, pcache0)[datakey(ino0, ino1), pe], ∀ key, ino. key ∈ evict(ino, pcache0) ∧ (∀ key. key ∈ pcache0 → key.data?) → key.data? ∧ key.ino ≠ ino, ∀ key. key ∈ pcache0[datakey(ino0, ino1), pe] → key.data? ⊦ ino = ino0, key.data? ∧ key.ino ≠ ino

Rule argument

spec: pcache-del-ops name: evict-rec-other seq: n ≠ ino ⊦ evict(n, pcache[datakey(ino, pageno), pe]) = evict(n, pcache)[datakey(ino, pageno), pe] subst: [pe, pageno, ino, pcache, n] → [pe, ino1, ino0, pcache0, ino] flags: <all paths>

Additional information

Used terms: pe, ino1, ino0, pcache0, ino