Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar3/nonterminSLASHCSRSLASHEx5

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, f₁(true))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

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

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

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

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

The set Q consists of the following terms:

f₁(x0)
if₃(true, x0, x1)
if₃(false, x0, x1)

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

F₁(X) -> F₁(true)
F₁(X) -> IF₃(X, c, f₁(true))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f₁(x0)
if₃(true, x0, x1)
if₃(false, x0, x1)

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

F₁(X) -> F₁(true)
F₁(X) -> IF₃(X, c, f₁(true))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f₁(x0)
if₃(true, x0, x1)
if₃(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 1 SCC with 1 less node.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

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

F₁(X) -> F₁(true)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

f₁(x0)
if₃(true, x0, x1)
if₃(false, x0, x1)

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