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:

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

Q is empty.


QTRS
  ↳ Overlay + Local Confluence

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

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS
  ↳ Overlay + Local Confluence
QTRS
      ↳ DependencyPairsProof

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

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)


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

IFFACT(x, true) → *1(x, fact(-(x, s(0))))
*1(x, s(y)) → +1(*(x, y), x)
+1(x, s(y)) → +1(x, y)
*1(x, s(y)) → *1(x, y)
FACT(x) → IFFACT(x, ge(x, s(s(0))))
-1(s(x), s(y)) → -1(x, y)
IFFACT(x, true) → FACT(-(x, s(0)))
IFFACT(x, true) → -1(x, s(0))
FACT(x) → GE(x, s(s(0)))
GE(s(x), s(y)) → GE(x, y)

The TRS R consists of the following rules:

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
QDP
          ↳ EdgeDeletionProof

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

IFFACT(x, true) → *1(x, fact(-(x, s(0))))
*1(x, s(y)) → +1(*(x, y), x)
+1(x, s(y)) → +1(x, y)
*1(x, s(y)) → *1(x, y)
FACT(x) → IFFACT(x, ge(x, s(s(0))))
-1(s(x), s(y)) → -1(x, y)
IFFACT(x, true) → FACT(-(x, s(0)))
IFFACT(x, true) → -1(x, s(0))
FACT(x) → GE(x, s(s(0)))
GE(s(x), s(y)) → GE(x, y)

The TRS R consists of the following rules:

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
QDP
              ↳ DependencyGraphProof

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

*1(x, s(y)) → +1(*(x, y), x)
IFFACT(x, true) → *1(x, fact(-(x, s(0))))
+1(x, s(y)) → +1(x, y)
*1(x, s(y)) → *1(x, y)
-1(s(x), s(y)) → -1(x, y)
FACT(x) → IFFACT(x, ge(x, s(s(0))))
IFFACT(x, true) → FACT(-(x, s(0)))
IFFACT(x, true) → -1(x, s(0))
FACT(x) → GE(x, s(s(0)))
GE(s(x), s(y)) → GE(x, y)

The TRS R consists of the following rules:

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 5 SCCs with 4 less nodes.

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
QDP
                    ↳ QDPOrderProof
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

-1(s(x), s(y)) → -1(x, y)

The TRS R consists of the following rules:

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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


The following pairs can be oriented strictly and are deleted.


-1(s(x), s(y)) → -1(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
-1(x1, x2)  =  x2
s(x1)  =  s(x1)

Recursive Path Order [2].
Precedence:
trivial


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
QDP
                    ↳ QDPOrderProof
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

GE(s(x), s(y)) → GE(x, y)

The TRS R consists of the following rules:

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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


The following pairs can be oriented strictly and are deleted.


GE(s(x), s(y)) → GE(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
GE(x1, x2)  =  x2
s(x1)  =  s(x1)

Recursive Path Order [2].
Precedence:
trivial


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP

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

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
QDP
                    ↳ QDPOrderProof
                  ↳ QDP
                  ↳ QDP

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

+1(x, s(y)) → +1(x, y)

The TRS R consists of the following rules:

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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


The following pairs can be oriented strictly and are deleted.


+1(x, s(y)) → +1(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
+1(x1, x2)  =  x2
s(x1)  =  s(x1)

Recursive Path Order [2].
Precedence:
trivial


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
                  ↳ QDP
                  ↳ QDP

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

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
QDP
                    ↳ QDPOrderProof
                  ↳ QDP

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

*1(x, s(y)) → *1(x, y)

The TRS R consists of the following rules:

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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


The following pairs can be oriented strictly and are deleted.


*1(x, s(y)) → *1(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
*1(x1, x2)  =  x2
s(x1)  =  s(x1)

Recursive Path Order [2].
Precedence:
trivial


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
                  ↳ QDP

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

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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

↳ QTRS
  ↳ Overlay + Local Confluence
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ EdgeDeletionProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
                  ↳ QDP
QDP

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

FACT(x) → IFFACT(x, ge(x, s(s(0))))
IFFACT(x, true) → FACT(-(x, s(0)))

The TRS R consists of the following rules:

+(x, 0) → x
+(x, s(y)) → s(+(x, y))
*(x, 0) → 0
*(x, s(y)) → +(*(x, y), x)
ge(x, 0) → true
ge(0, s(y)) → false
ge(s(x), s(y)) → ge(x, y)
-(x, 0) → x
-(s(x), s(y)) → -(x, y)
fact(x) → iffact(x, ge(x, s(s(0))))
iffact(x, true) → *(x, fact(-(x, s(0))))
iffact(x, false) → s(0)

The set Q consists of the following terms:

+(x0, 0)
+(x0, s(x1))
*(x0, 0)
*(x0, s(x1))
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
-(x0, 0)
-(s(x0), s(x1))
fact(x0)
iffact(x0, true)
iffact(x0, false)

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