Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/nonterminSLASHCSRSLASHExIntrod

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:

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

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:

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

The set Q consists of the following terms:

fact₁(x0)
add₂(0, x0)
add₂(s₁(x0), x1)
prod₂(0, x0)
prod₂(s₁(x0), x1)
if₃(true, x0, x1)
if₃(false, x0, x1)
zero₁(0)
zero₁(s₁(x0))
p₁(s₁(x0))

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

FACT₁(X) -> ZERO₁(X)
FACT₁(X) -> PROD₂(X, fact₁(p₁(X)))
PROD₂(s₁(X), Y) -> ADD₂(Y, prod₂(X, Y))
ADD₂(s₁(X), Y) -> ADD₂(X, Y)
FACT₁(X) -> P₁(X)
FACT₁(X) -> IF₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
FACT₁(X) -> FACT₁(p₁(X))
PROD₂(s₁(X), Y) -> PROD₂(X, Y)

The TRS R consists of the following rules:

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

The set Q consists of the following terms:

fact₁(x0)
add₂(0, x0)
add₂(s₁(x0), x1)
prod₂(0, x0)
prod₂(s₁(x0), x1)
if₃(true, x0, x1)
if₃(false, x0, x1)
zero₁(0)
zero₁(s₁(x0))
p₁(s₁(x0))

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:

FACT₁(X) -> ZERO₁(X)
FACT₁(X) -> PROD₂(X, fact₁(p₁(X)))
PROD₂(s₁(X), Y) -> ADD₂(Y, prod₂(X, Y))
ADD₂(s₁(X), Y) -> ADD₂(X, Y)
FACT₁(X) -> P₁(X)
FACT₁(X) -> IF₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
FACT₁(X) -> FACT₁(p₁(X))
PROD₂(s₁(X), Y) -> PROD₂(X, Y)

The TRS R consists of the following rules:

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

The set Q consists of the following terms:

fact₁(x0)
add₂(0, x0)
add₂(s₁(x0), x1)
prod₂(0, x0)
prod₂(s₁(x0), x1)
if₃(true, x0, x1)
if₃(false, x0, x1)
zero₁(0)
zero₁(s₁(x0))
p₁(s₁(x0))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 5 less nodes.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

              ↳ QDP

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

ADD₂(s₁(X), Y) -> ADD₂(X, Y)

The TRS R consists of the following rules:

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

The set Q consists of the following terms:

fact₁(x0)
add₂(0, x0)
add₂(s₁(x0), x1)
prod₂(0, x0)
prod₂(s₁(x0), x1)
if₃(true, x0, x1)
if₃(false, x0, x1)
zero₁(0)
zero₁(s₁(x0))
p₁(s₁(x0))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be strictly oriented and are deleted.

ADD₂(s₁(X), Y) -> ADD₂(X, Y)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
ADD₂(x1, x2) = ADD₁(x1)
s₁(x1) = s₁(x1)

Lexicographic Path Order [19].
Precedence:

[ADD₁, s_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

              ↳ QDP

              ↳ QDP

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

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

The set Q consists of the following terms:

fact₁(x0)
add₂(0, x0)
add₂(s₁(x0), x1)
prod₂(0, x0)
prod₂(s₁(x0), x1)
if₃(true, x0, x1)
if₃(false, x0, x1)
zero₁(0)
zero₁(s₁(x0))
p₁(s₁(x0))

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

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

PROD₂(s₁(X), Y) -> PROD₂(X, Y)

The TRS R consists of the following rules:

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

The set Q consists of the following terms:

fact₁(x0)
add₂(0, x0)
add₂(s₁(x0), x1)
prod₂(0, x0)
prod₂(s₁(x0), x1)
if₃(true, x0, x1)
if₃(false, x0, x1)
zero₁(0)
zero₁(s₁(x0))
p₁(s₁(x0))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be strictly oriented and are deleted.

PROD₂(s₁(X), Y) -> PROD₂(X, Y)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
PROD₂(x1, x2) = PROD₁(x1)
s₁(x1) = s₁(x1)

Lexicographic Path Order [19].
Precedence:

[PROD₁, s_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

              ↳ QDP

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

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

The set Q consists of the following terms:

fact₁(x0)
add₂(0, x0)
add₂(s₁(x0), x1)
prod₂(0, x0)
prod₂(s₁(x0), x1)
if₃(true, x0, x1)
if₃(false, x0, x1)
zero₁(0)
zero₁(s₁(x0))
p₁(s₁(x0))

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

FACT₁(X) -> FACT₁(p₁(X))

The TRS R consists of the following rules:

fact₁(X) -> if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(add₂(X, Y))
prod₂(0, X) -> 0
prod₂(s₁(X), Y) -> add₂(Y, prod₂(X, Y))
if₃(true, X, Y) -> X
if₃(false, X, Y) -> Y
zero₁(0) -> true
zero₁(s₁(X)) -> false
p₁(s₁(X)) -> X

The set Q consists of the following terms:

fact₁(x0)
add₂(0, x0)
add₂(s₁(x0), x1)
prod₂(0, x0)
prod₂(s₁(x0), x1)
if₃(true, x0, x1)
if₃(false, x0, x1)
zero₁(0)
zero₁(s₁(x0))
p₁(s₁(x0))

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