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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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(*2(x, y), z) -> *12(y, z)
PROD1(app2(l1, l2)) -> PROD1(l2)
SUM1(app2(l1, l2)) -> SUM1(l1)
*12(s1(x), s1(y)) -> +12(*2(x, y), +2(x, y))
SUM1(app2(l1, l2)) -> +12(sum1(l1), sum1(l2))
SUM1(app2(l1, l2)) -> SUM1(l2)
*12(s1(x), s1(y)) -> *12(x, y)
+12(+2(x, y), z) -> +12(x, +2(y, z))
*12(*2(x, y), z) -> *12(x, *2(y, z))
SUM1(cons2(x, l)) -> SUM1(l)
+12(s1(x), s1(y)) -> +12(x, y)
PROD1(cons2(x, l)) -> *12(x, prod1(l))
PROD1(app2(l1, l2)) -> *12(prod1(l1), prod1(l2))
PROD1(cons2(x, l)) -> PROD1(l)
+12(+2(x, y), z) -> +12(y, z)
PROD1(app2(l1, l2)) -> PROD1(l1)
*12(s1(x), s1(y)) -> +12(x, y)
APP2(cons2(x, l1), l2) -> APP2(l1, l2)

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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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(*2(x, y), z) -> *12(y, z)
PROD1(app2(l1, l2)) -> PROD1(l2)
SUM1(app2(l1, l2)) -> SUM1(l1)
*12(s1(x), s1(y)) -> +12(*2(x, y), +2(x, y))
SUM1(app2(l1, l2)) -> +12(sum1(l1), sum1(l2))
SUM1(app2(l1, l2)) -> SUM1(l2)
*12(s1(x), s1(y)) -> *12(x, y)
+12(+2(x, y), z) -> +12(x, +2(y, z))
*12(*2(x, y), z) -> *12(x, *2(y, z))
SUM1(cons2(x, l)) -> SUM1(l)
+12(s1(x), s1(y)) -> +12(x, y)
PROD1(cons2(x, l)) -> *12(x, prod1(l))
PROD1(app2(l1, l2)) -> *12(prod1(l1), prod1(l2))
PROD1(cons2(x, l)) -> PROD1(l)
+12(+2(x, y), z) -> +12(y, z)
PROD1(app2(l1, l2)) -> PROD1(l1)
*12(s1(x), s1(y)) -> +12(x, y)
APP2(cons2(x, l1), l2) -> APP2(l1, l2)

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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 5 SCCs with 6 less nodes.

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

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

APP2(cons2(x, l1), l2) -> APP2(l1, l2)

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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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.


APP2(cons2(x, l1), l2) -> APP2(l1, l2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(APP2(x1, x2)) = x1   
POL(cons2(x1, x2)) = 1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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:

+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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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
          ↳ 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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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)
+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)) = 1 + x1 + x2   
POL(+12(x1, x2)) = x1   
POL(0) = 0   
POL(s1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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:

+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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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
          ↳ QDP

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

SUM1(app2(l1, l2)) -> SUM1(l1)
SUM1(app2(l1, l2)) -> SUM1(l2)
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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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(app2(l1, l2)) -> SUM1(l1)
SUM1(app2(l1, l2)) -> SUM1(l2)
SUM1(cons2(x, l)) -> SUM1(l)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(SUM1(x1)) = x1   
POL(app2(x1, x2)) = 1 + x1 + x2   
POL(cons2(x1, x2)) = 1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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:

+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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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
          ↳ 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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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(s1(x), s1(y)) -> *12(x, y)
*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)) = 1 + x1 + x2   
POL(*12(x1, x2)) = x1   
POL(+2(x1, x2)) = 0   
POL(0) = 1   
POL(s1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ 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:

+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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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
          ↳ QDP
            ↳ QDPOrderProof

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

PROD1(app2(l1, l2)) -> PROD1(l2)
PROD1(cons2(x, l)) -> PROD1(l)
PROD1(app2(l1, l2)) -> PROD1(l1)

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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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(app2(l1, l2)) -> PROD1(l2)
PROD1(cons2(x, l)) -> PROD1(l)
PROD1(app2(l1, l2)) -> PROD1(l1)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROD1(x1)) = x1   
POL(app2(x1, x2)) = 1 + x1 + x2   
POL(cons2(x1, x2)) = 1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ 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))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 0
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> s1(0)
prod1(cons2(x, l)) -> *2(x, prod1(l))
prod1(app2(l1, l2)) -> *2(prod1(l1), prod1(l2))

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.