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:

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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:

-12(I1(x), I1(y)) -> O11(-2(x, y))
+12(O1(x), O1(y)) -> O11(+2(x, y))
+12(O1(x), I1(y)) -> +12(x, y)
+12(I1(x), O1(y)) -> +12(x, y)
-12(O1(x), I1(y)) -> -12(-2(x, y), I1(1))
+12(I1(x), I1(y)) -> +12(x, y)
*12(O1(x), y) -> *12(x, y)
-12(O1(x), O1(y)) -> O11(-2(x, y))
-12(I1(x), I1(y)) -> -12(x, y)
*12(I1(x), y) -> *12(x, y)
-12(I1(x), O1(y)) -> -12(x, y)
-12(O1(x), I1(y)) -> -12(x, y)
+12(I1(x), I1(y)) -> O11(+2(+2(x, y), I1(0)))
-12(O1(x), O1(y)) -> -12(x, y)
*12(I1(x), y) -> +12(O1(*2(x, y)), y)
+12(I1(x), I1(y)) -> +12(+2(x, y), I1(0))
+12(O1(x), O1(y)) -> +12(x, y)
*12(I1(x), y) -> O11(*2(x, y))
*12(O1(x), y) -> O11(*2(x, y))

The TRS R consists of the following rules:

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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:

-12(I1(x), I1(y)) -> O11(-2(x, y))
+12(O1(x), O1(y)) -> O11(+2(x, y))
+12(O1(x), I1(y)) -> +12(x, y)
+12(I1(x), O1(y)) -> +12(x, y)
-12(O1(x), I1(y)) -> -12(-2(x, y), I1(1))
+12(I1(x), I1(y)) -> +12(x, y)
*12(O1(x), y) -> *12(x, y)
-12(O1(x), O1(y)) -> O11(-2(x, y))
-12(I1(x), I1(y)) -> -12(x, y)
*12(I1(x), y) -> *12(x, y)
-12(I1(x), O1(y)) -> -12(x, y)
-12(O1(x), I1(y)) -> -12(x, y)
+12(I1(x), I1(y)) -> O11(+2(+2(x, y), I1(0)))
-12(O1(x), O1(y)) -> -12(x, y)
*12(I1(x), y) -> +12(O1(*2(x, y)), y)
+12(I1(x), I1(y)) -> +12(+2(x, y), I1(0))
+12(O1(x), O1(y)) -> +12(x, y)
*12(I1(x), y) -> O11(*2(x, y))
*12(O1(x), y) -> O11(*2(x, y))

The TRS R consists of the following rules:

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 3 SCCs with 7 less nodes.

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

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

-12(I1(x), I1(y)) -> -12(x, y)
-12(I1(x), O1(y)) -> -12(x, y)
-12(O1(x), I1(y)) -> -12(x, y)
-12(O1(x), O1(y)) -> -12(x, y)
-12(O1(x), I1(y)) -> -12(-2(x, y), I1(1))

The TRS R consists of the following rules:

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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(I1(x), O1(y)) -> -12(x, y)
-12(O1(x), O1(y)) -> -12(x, y)
The remaining pairs can at least be oriented weakly.

-12(I1(x), I1(y)) -> -12(x, y)
-12(O1(x), I1(y)) -> -12(x, y)
-12(O1(x), I1(y)) -> -12(-2(x, y), I1(1))
Used ordering: Polynomial interpretation [21]:

POL(-2(x1, x2)) = 0   
POL(-12(x1, x2)) = x2   
POL(0) = 0   
POL(1) = 0   
POL(I1(x1)) = 2·x1   
POL(O1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



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

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

-12(I1(x), I1(y)) -> -12(x, y)
-12(O1(x), I1(y)) -> -12(x, y)
-12(O1(x), I1(y)) -> -12(-2(x, y), I1(1))

The TRS R consists of the following rules:

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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(I1(x), I1(y)) -> -12(x, y)
-12(O1(x), I1(y)) -> -12(x, y)
The remaining pairs can at least be oriented weakly.

-12(O1(x), I1(y)) -> -12(-2(x, y), I1(1))
Used ordering: Polynomial interpretation [21]:

POL(-2(x1, x2)) = 0   
POL(-12(x1, x2)) = 2·x2   
POL(0) = 0   
POL(1) = 0   
POL(I1(x1)) = 1 + x1   
POL(O1(x1)) = 0   

The following usable rules [14] were oriented: none



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

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

-12(O1(x), I1(y)) -> -12(-2(x, y), I1(1))

The TRS R consists of the following rules:

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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(O1(x), I1(y)) -> -12(-2(x, y), I1(1))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(-2(x1, x2)) = x1 + x2   
POL(-12(x1, x2)) = 3·x1 + 2·x2   
POL(0) = 3   
POL(1) = 0   
POL(I1(x1)) = 1 + 2·x1   
POL(O1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented:

O1(0) -> 0
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(0, x) -> 0
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(x, 0) -> x
-2(I1(x), I1(y)) -> O1(-2(x, y))



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

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

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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

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

+12(I1(x), O1(y)) -> +12(x, y)
+12(O1(x), I1(y)) -> +12(x, y)
+12(I1(x), I1(y)) -> +12(+2(x, y), I1(0))
+12(I1(x), I1(y)) -> +12(x, y)
+12(O1(x), O1(y)) -> +12(x, y)

The TRS R consists of the following rules:

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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(I1(x), O1(y)) -> +12(x, y)
+12(O1(x), I1(y)) -> +12(x, y)
+12(I1(x), I1(y)) -> +12(+2(x, y), I1(0))
+12(I1(x), I1(y)) -> +12(x, y)
+12(O1(x), O1(y)) -> +12(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(+2(x1, x2)) = 2 + x1 + x2   
POL(+12(x1, x2)) = 3·x1 + 3·x2   
POL(0) = 0   
POL(I1(x1)) = 3 + x1   
POL(O1(x1)) = 1 + x1   

The following usable rules [14] were oriented:

O1(0) -> 0
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
+2(x, 0) -> x
+2(0, x) -> x
+2(O1(x), I1(y)) -> I1(+2(x, y))



↳ 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:

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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

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

*12(I1(x), y) -> *12(x, y)
*12(O1(x), y) -> *12(x, y)

The TRS R consists of the following rules:

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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(I1(x), y) -> *12(x, y)
*12(O1(x), y) -> *12(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(*12(x1, x2)) = 3·x1   
POL(I1(x1)) = 3 + x1   
POL(O1(x1)) = 3 + 2·x1   

The following usable rules [14] were oriented: none



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

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

O1(0) -> 0
+2(0, x) -> x
+2(x, 0) -> x
+2(O1(x), O1(y)) -> O1(+2(x, y))
+2(O1(x), I1(y)) -> I1(+2(x, y))
+2(I1(x), O1(y)) -> I1(+2(x, y))
+2(I1(x), I1(y)) -> O1(+2(+2(x, y), I1(0)))
*2(0, x) -> 0
*2(x, 0) -> 0
*2(O1(x), y) -> O1(*2(x, y))
*2(I1(x), y) -> +2(O1(*2(x, y)), y)
-2(x, 0) -> x
-2(0, x) -> 0
-2(O1(x), O1(y)) -> O1(-2(x, y))
-2(O1(x), I1(y)) -> I1(-2(-2(x, y), I1(1)))
-2(I1(x), O1(y)) -> I1(-2(x, y))
-2(I1(x), I1(y)) -> O1(-2(x, y))

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.