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:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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(nil) -> 011(#)
*12(*2(x, y), z) -> *12(y, z)
PROD1(app2(l1, l2)) -> PROD1(l2)
SUM1(app2(l1, l2)) -> +12(sum1(l1), sum1(l2))
*12(x, +2(y, z)) -> *12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))
+12(11(x), 11(y)) -> 011(+2(+2(x, y), 11(#)))
+12(01(x), 01(y)) -> +12(x, y)
*12(x, +2(y, z)) -> +12(*2(x, y), *2(x, z))
+12(+2(x, y), z) -> +12(y, z)
PROD1(app2(l1, l2)) -> PROD1(l1)
*12(11(x), y) -> 011(*2(x, y))
*12(11(x), y) -> *12(x, y)
+12(01(x), 01(y)) -> 011(+2(x, y))
*12(11(x), y) -> +12(01(*2(x, y)), y)
APP2(cons2(x, l1), l2) -> APP2(l1, l2)
SUM1(cons2(x, l)) -> +12(x, sum1(l))
+12(11(x), 11(y)) -> +12(x, y)
SUM1(app2(l1, l2)) -> SUM1(l1)
SUM1(app2(l1, l2)) -> SUM1(l2)
+12(+2(x, y), z) -> +12(x, +2(y, z))
SUM1(cons2(x, l)) -> SUM1(l)
PROD1(cons2(x, l)) -> *12(x, prod1(l))
PROD1(app2(l1, l2)) -> *12(prod1(l1), prod1(l2))
PROD1(cons2(x, l)) -> PROD1(l)
*12(x, +2(y, z)) -> *12(x, z)
*12(01(x), y) -> 011(*2(x, y))
*12(01(x), y) -> *12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))

The TRS R consists of the following rules:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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(nil) -> 011(#)
*12(*2(x, y), z) -> *12(y, z)
PROD1(app2(l1, l2)) -> PROD1(l2)
SUM1(app2(l1, l2)) -> +12(sum1(l1), sum1(l2))
*12(x, +2(y, z)) -> *12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))
+12(11(x), 11(y)) -> 011(+2(+2(x, y), 11(#)))
+12(01(x), 01(y)) -> +12(x, y)
*12(x, +2(y, z)) -> +12(*2(x, y), *2(x, z))
+12(+2(x, y), z) -> +12(y, z)
PROD1(app2(l1, l2)) -> PROD1(l1)
*12(11(x), y) -> 011(*2(x, y))
*12(11(x), y) -> *12(x, y)
+12(01(x), 01(y)) -> 011(+2(x, y))
*12(11(x), y) -> +12(01(*2(x, y)), y)
APP2(cons2(x, l1), l2) -> APP2(l1, l2)
SUM1(cons2(x, l)) -> +12(x, sum1(l))
+12(11(x), 11(y)) -> +12(x, y)
SUM1(app2(l1, l2)) -> SUM1(l1)
SUM1(app2(l1, l2)) -> SUM1(l2)
+12(+2(x, y), z) -> +12(x, +2(y, z))
SUM1(cons2(x, l)) -> SUM1(l)
PROD1(cons2(x, l)) -> *12(x, prod1(l))
PROD1(app2(l1, l2)) -> *12(prod1(l1), prod1(l2))
PROD1(cons2(x, l)) -> PROD1(l)
*12(x, +2(y, z)) -> *12(x, z)
*12(01(x), y) -> 011(*2(x, y))
*12(01(x), y) -> *12(x, y)
+12(11(x), 01(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))

The TRS R consists of the following rules:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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 11 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:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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 Order [17,21] with Interpretation:

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


POL( APP2(x1, x2) ) = x1 + 3



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:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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(01(x), 01(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
+12(11(x), 01(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(x, +2(y, z))
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))

The TRS R consists of the following rules:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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(11(x), 11(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
+12(11(x), 01(y)) -> +12(x, y)
+12(01(x), 11(y)) -> +12(x, y)
+12(11(x), 11(y)) -> +12(+2(x, y), 11(#))
The remaining pairs can at least be oriented weakly.

+12(01(x), 01(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(x, +2(y, z))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( 01(x1) ) = x1


POL( 11(x1) ) = x1 + 3


POL( +12(x1, x2) ) = x1 + x2 + 3


POL( +2(x1, x2) ) = x1 + x2 + 2


POL( # ) = 0



The following usable rules [14] were oriented:

+2(01(x), 11(y)) -> 11(+2(x, y))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(x, #) -> x
01(#) -> #
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(#, x) -> x
+2(11(x), 01(y)) -> 11(+2(x, y))



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

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

+12(01(x), 01(y)) -> +12(x, y)
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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(01(x), 01(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 Order [17,21] with Interpretation:

POL( 01(x1) ) = 2x1 + 2


POL( 11(x1) ) = max{0, 2x1 - 3}


POL( +12(x1, x2) ) = x1 + 3


POL( +2(x1, x2) ) = x1 + 1


POL( # ) = 3



The following usable rules [14] were oriented: none



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

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

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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 Order [17,21] with Interpretation:

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


POL( app2(x1, x2) ) = 3x1 + 2x2 + 3


POL( SUM1(x1) ) = x1 + 2



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:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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(x, +2(y, z)) -> *12(x, z)
*12(11(x), y) -> *12(x, y)
*12(x, +2(y, z)) -> *12(x, y)
*12(01(x), y) -> *12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))

The TRS R consists of the following rules:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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(11(x), y) -> *12(x, y)
*12(01(x), y) -> *12(x, y)
*12(*2(x, y), z) -> *12(x, *2(y, z))
The remaining pairs can at least be oriented weakly.

*12(x, +2(y, z)) -> *12(x, z)
*12(x, +2(y, z)) -> *12(x, y)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( *12(x1, x2) ) = 2x1


POL( 01(x1) ) = 2x1 + 2


POL( 11(x1) ) = 2x1 + 2


POL( # ) = 2


POL( +2(x1, x2) ) = 3


POL( *2(x1, x2) ) = 2x1 + 2x2 + 1



The following usable rules [14] were oriented: none



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

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

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

The TRS R consists of the following rules:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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(x, +2(y, z)) -> *12(x, z)
*12(x, +2(y, z)) -> *12(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( *12(x1, x2) ) = 3x1 + x2 + 3


POL( +2(x1, x2) ) = 2x1 + 2x2 + 1



The following usable rules [14] were oriented: none



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

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

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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 Order [17,21] with Interpretation:

POL( cons2(x1, x2) ) = 3x1 + 2x2 + 1


POL( PROD1(x1) ) = x1 + 3


POL( app2(x1, x2) ) = 2x1 + 2x2 + 1



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:

01(#) -> #
+2(x, #) -> x
+2(#, x) -> x
+2(01(x), 01(y)) -> 01(+2(x, y))
+2(01(x), 11(y)) -> 11(+2(x, y))
+2(11(x), 01(y)) -> 11(+2(x, y))
+2(11(x), 11(y)) -> 01(+2(+2(x, y), 11(#)))
+2(+2(x, y), z) -> +2(x, +2(y, z))
*2(#, x) -> #
*2(01(x), y) -> 01(*2(x, y))
*2(11(x), y) -> +2(01(*2(x, y)), y)
*2(*2(x, y), z) -> *2(x, *2(y, z))
*2(x, +2(y, z)) -> +2(*2(x, y), *2(x, z))
app2(nil, l) -> l
app2(cons2(x, l1), l2) -> cons2(x, app2(l1, l2))
sum1(nil) -> 01(#)
sum1(cons2(x, l)) -> +2(x, sum1(l))
sum1(app2(l1, l2)) -> +2(sum1(l1), sum1(l2))
prod1(nil) -> 11(#)
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.