Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar14/TRCSRSLASHEx5_Zan97

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

f₁(X) -> if₃(X, c, n__f₁(true))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> activate₁(Y)
f₁(X) -> n__f₁(X)
activate₁(n__f₁(X)) -> f₁(X)
activate₁(X) -> X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

f₁(X) -> if₃(X, c, n__f₁(true))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> activate₁(Y)
f₁(X) -> n__f₁(X)
activate₁(n__f₁(X)) -> f₁(X)
activate₁(X) -> X

Q is empty.

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

ACTIVATE₁(n__f₁(X)) -> F₁(X)
IF₃(false, X, Y) -> ACTIVATE₁(Y)
F₁(X) -> IF₃(X, c, n__f₁(true))

The TRS R consists of the following rules:

f₁(X) -> if₃(X, c, n__f₁(true))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> activate₁(Y)
f₁(X) -> n__f₁(X)
activate₁(n__f₁(X)) -> f₁(X)
activate₁(X) -> X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE₁(n__f₁(X)) -> F₁(X)
IF₃(false, X, Y) -> ACTIVATE₁(Y)
F₁(X) -> IF₃(X, c, n__f₁(true))

The TRS R consists of the following rules:

f₁(X) -> if₃(X, c, n__f₁(true))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> activate₁(Y)
f₁(X) -> n__f₁(X)
activate₁(n__f₁(X)) -> f₁(X)
activate₁(X) -> X

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