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, s1]


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, s1]


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.