Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

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:

SUM1(cons2(x, l)) -> +12(x, sum1(l))
+12(s1(x), s1(y)) -> +12(x, y)
PROD1(cons2(x, l)) -> *12(x, prod1(l))
*12(*2(x, y), z) -> *12(y, z)
PROD1(cons2(x, l)) -> PROD1(l)
+12(+2(x, y), z) -> +12(y, z)
*12(s1(x), s1(y)) -> +12(*2(x, y), +2(x, y))
*12(s1(x), s1(y)) -> +12(x, y)
*12(s1(x), s1(y)) -> *12(x, y)
SUM1(cons2(x, l)) -> SUM1(l)
*12(*2(x, y), z) -> *12(x, *2(y, z))
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

SUM1(cons2(x, l)) -> +12(x, sum1(l))
+12(s1(x), s1(y)) -> +12(x, y)
PROD1(cons2(x, l)) -> *12(x, prod1(l))
*12(*2(x, y), z) -> *12(y, z)
PROD1(cons2(x, l)) -> PROD1(l)
+12(+2(x, y), z) -> +12(y, z)
*12(s1(x), s1(y)) -> +12(*2(x, y), +2(x, y))
*12(s1(x), s1(y)) -> +12(x, y)
*12(s1(x), s1(y)) -> *12(x, y)
SUM1(cons2(x, l)) -> SUM1(l)
*12(*2(x, y), z) -> *12(x, *2(y, z))
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

+12(s1(x), s1(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

Q is empty.
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.


+12(s1(x), s1(y)) -> +12(x, y)
The remaining pairs can at least be oriented weakly.

+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))
Used ordering: Polynomial interpretation [21]:

POL(+2(x1, x2)) = 2·x1 + 2·x2   
POL(+12(x1, x2)) = 2·x1   
POL(0) = 0   
POL(s1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

Q is empty.
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.


+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(+2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(+12(x1, x2)) = 2·x1   
POL(0) = 0   
POL(s1(x1)) = 0   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

SUM1(cons2(x, l)) -> SUM1(l)

The TRS R consists of the following rules:

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

Q is empty.
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.


SUM1(cons2(x, l)) -> SUM1(l)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(SUM1(x1)) = 2·x1   
POL(cons2(x1, x2)) = 1 + 2·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

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

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

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

*12(*2(x, y), z) -> *12(y, z)
*12(s1(x), s1(y)) -> *12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))

The TRS R consists of the following rules:

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

Q is empty.
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.


*12(s1(x), s1(y)) -> *12(x, y)
The remaining pairs can at least be oriented weakly.

*12(*2(x, y), z) -> *12(y, z)
*12(*2(x, y), z) -> *12(x, *2(y, z))
Used ordering: Polynomial interpretation [21]:

POL(*2(x1, x2)) = 2·x1 + 2·x2   
POL(*12(x1, x2)) = 2·x1   
POL(+2(x1, x2)) = 0   
POL(0) = 0   
POL(s1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP

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

*12(*2(x, y), z) -> *12(y, z)
*12(*2(x, y), z) -> *12(x, *2(y, z))

The TRS R consists of the following rules:

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

Q is empty.
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.


*12(*2(x, y), z) -> *12(y, z)
*12(*2(x, y), z) -> *12(x, *2(y, z))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(*2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(*12(x1, x2)) = 2·x1   
POL(+2(x1, x2)) = 0   
POL(0) = 0   
POL(s1(x1)) = 0   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP

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

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof

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

PROD1(cons2(x, l)) -> PROD1(l)

The TRS R consists of the following rules:

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

Q is empty.
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.


PROD1(cons2(x, l)) -> PROD1(l)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROD1(x1)) = 2·x1   
POL(cons2(x1, x2)) = 1 + 2·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof

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

+2(x, 0) -> x
+2(0, x) -> x
+2(s1(x), s1(y)) -> s1(s1(+2(x, y)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(x, 0) -> 0
*2(0, x) -> 0
*2(s1(x), s1(y)) -> s1(+2(*2(x, y), +2(x, y)))
*2(*2(x, y), z) -> *2(x, *2(y, z))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))

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