Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

app2(app2(, x), x) -> e
app2(app2(, e), x) -> x
app2(app2(, x), app2(app2(., x), y)) -> y
app2(app2(, app2(app2(/, x), y)), x) -> y
app2(app2(/, x), x) -> e
app2(app2(/, x), e) -> x
app2(app2(/, app2(app2(., y), x)), x) -> y
app2(app2(/, x), app2(app2(, y), x)) -> y
app2(app2(., e), x) -> x
app2(app2(., x), e) -> x
app2(app2(., x), app2(app2(, x), y)) -> y
app2(app2(., app2(app2(/, y), x)), x) -> y

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

app2(app2(, x), x) -> e
app2(app2(, e), x) -> x
app2(app2(, x), app2(app2(., x), y)) -> y
app2(app2(, app2(app2(/, x), y)), x) -> y
app2(app2(/, x), x) -> e
app2(app2(/, x), e) -> x
app2(app2(/, app2(app2(., y), x)), x) -> y
app2(app2(/, x), app2(app2(, y), x)) -> y
app2(app2(., e), x) -> x
app2(app2(., x), e) -> x
app2(app2(., x), app2(app2(, x), y)) -> y
app2(app2(., app2(app2(/, y), x)), x) -> y

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

app2(app2(, x), x) -> e
app2(app2(, e), x) -> x
app2(app2(, x), app2(app2(., x), y)) -> y
app2(app2(, app2(app2(/, x), y)), x) -> y
app2(app2(/, x), x) -> e
app2(app2(/, x), e) -> x
app2(app2(/, app2(app2(., y), x)), x) -> y
app2(app2(/, x), app2(app2(, y), x)) -> y
app2(app2(., e), x) -> x
app2(app2(., x), e) -> x
app2(app2(., x), app2(app2(, x), y)) -> y
app2(app2(., app2(app2(/, y), x)), x) -> y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

app2(app2(, x), x) -> e
app2(app2(, e), x) -> x
app2(app2(, x), app2(app2(., x), y)) -> y
app2(app2(, app2(app2(/, x), y)), x) -> y
app2(app2(/, x), x) -> e
app2(app2(/, x), e) -> x
app2(app2(/, app2(app2(., y), x)), x) -> y
app2(app2(/, x), app2(app2(, y), x)) -> y
app2(app2(., e), x) -> x
app2(app2(., x), e) -> x
app2(app2(., x), app2(app2(, x), y)) -> y
app2(app2(., app2(app2(/, y), x)), x) -> y

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.