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(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

*2(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))

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(*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(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof

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(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))

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)
The remaining pairs can at least be oriented weakly.

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

POL( 1 ) = 2


POL( 0 ) = 0


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


POL( i1(x1) ) = 2x1 + 2


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



The following usable rules [14] were oriented:

*2(x, 0) -> 0
*2(i1(x), x) -> 1
*2(1, y) -> y
*2(*2(x, y), z) -> *2(x, *2(y, z))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

*2(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))

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(x, *2(y, z))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( 1 ) = 2


POL( 0 ) = max{0, -1}


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


POL( i1(x1) ) = max{0, 2x1 - 1}


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



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ PisEmptyProof

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

*2(i1(x), x) -> 1
*2(1, y) -> y
*2(x, 0) -> 0
*2(*2(x, y), z) -> *2(x, *2(y, z))

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.