Term Rewriting System R:
[x, y, z]
*(*(x, y), z) -> *(x, *(y, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, +(y, f(z))) -> *(g(x, z), +(y, y))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

*'(*(x, y), z) -> *'(x, *(y, z))
*'(*(x, y), z) -> *'(y, z)
*'(+(x, y), z) -> *'(x, z)
*'(+(x, y), z) -> *'(y, z)
*'(x, +(y, f(z))) -> *'(g(x, z), +(y, y))

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pair:

*'(x, +(y, f(z))) -> *'(g(x, z), +(y, y))


Rules:


*(*(x, y), z) -> *(x, *(y, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, +(y, f(z))) -> *(g(x, z), +(y, y))





On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

*'(x, +(y, f(z))) -> *'(g(x, z), +(y, y))
one new Dependency Pair is created:

*'(g(x'', z'''), +(y0, f(z''))) -> *'(g(g(x'', z'''), z''), +(y0, y0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 3
             ↳Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pair:

*'(g(x'', z'''), +(y0, f(z''))) -> *'(g(g(x'', z'''), z''), +(y0, y0))


Rules:


*(*(x, y), z) -> *(x, *(y, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, +(y, f(z))) -> *(g(x, z), +(y, y))





On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

*'(g(x'', z'''), +(y0, f(z''))) -> *'(g(g(x'', z'''), z''), +(y0, y0))
one new Dependency Pair is created:

*'(g(g(x'''', z''''''), z'''''), +(y00, f(z''1))) -> *'(g(g(g(x'''', z''''''), z'''''), z''1), +(y00, y00))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 3
             ↳Inst
             ...
               →DP Problem 4
                 ↳Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pair:

*'(g(g(x'''', z''''''), z'''''), +(y00, f(z''1))) -> *'(g(g(g(x'''', z''''''), z'''''), z''1), +(y00, y00))


Rules:


*(*(x, y), z) -> *(x, *(y, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, +(y, f(z))) -> *(g(x, z), +(y, y))





On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

*'(g(g(x'''', z''''''), z'''''), +(y00, f(z''1))) -> *'(g(g(g(x'''', z''''''), z'''''), z''1), +(y00, y00))
one new Dependency Pair is created:

*'(g(g(g(x'''''', z''''''''), z''''''0), z'''''1), +(y000, f(z''1''))) -> *'(g(g(g(g(x'''''', z''''''''), z''''''0), z'''''1), z''1''), +(y000, y000))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 3
             ↳Inst
             ...
               →DP Problem 5
                 ↳Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pair:

*'(g(g(g(x'''''', z''''''''), z''''''0), z'''''1), +(y000, f(z''1''))) -> *'(g(g(g(g(x'''''', z''''''''), z''''''0), z'''''1), z''1''), +(y000, y000))


Rules:


*(*(x, y), z) -> *(x, *(y, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, +(y, f(z))) -> *(g(x, z), +(y, y))





On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

*'(g(g(g(x'''''', z''''''''), z''''''0), z'''''1), +(y000, f(z''1''))) -> *'(g(g(g(g(x'''''', z''''''''), z''''''0), z'''''1), z''1''), +(y000, y000))
one new Dependency Pair is created:

*'(g(g(g(g(x'''''''', z'''''''''''), z''''''''''), z''''''0''), z'''''1''), +(y0000, f(z''1''''))) -> *'(g(g(g(g(g(x'''''''', z'''''''''''), z''''''''''), z''''''0''), z'''''1''), z''1''''), +(y0000, y0000))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 3
             ↳Inst
             ...
               →DP Problem 6
                 ↳Instantiation Transformation
       →DP Problem 2
         ↳Remaining


Dependency Pair:

*'(g(g(g(g(x'''''''', z'''''''''''), z''''''''''), z''''''0''), z'''''1''), +(y0000, f(z''1''''))) -> *'(g(g(g(g(g(x'''''''', z'''''''''''), z''''''''''), z''''''0''), z'''''1''), z''1''''), +(y0000, y0000))


Rules:


*(*(x, y), z) -> *(x, *(y, z))
*(+(x, y), z) -> +(*(x, z), *(y, z))
*(x, +(y, f(z))) -> *(g(x, z), +(y, y))





On this DP problem, an Instantiation SCC transformation can be performed.
As a result of transforming the rule

*'(g(g(g(g(x'''''''', z'''''''''''), z''''''''''), z''''''0''), z'''''1''), +(y0000, f(z''1''''))) -> *'(g(g(g(g(g(x'''''''', z'''''''''''), z''''''''''), z''''''0''), z'''''1''), z''1''''), +(y0000, y0000))
one new Dependency Pair is created:

*'(g(g(g(g(g(x'''''''''', z''''''''''''''), z'''''''''''''), z''''''''''1), z''''''0''''), z'''''1''''), +(y00000, f(z''1''''''))) -> *'(g(g(g(g(g(g(x'''''''''', z''''''''''''''), z'''''''''''''), z''''''''''1), z''''''0''''), z'''''1''''), z''1''''''), +(y00000, y00000))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:


   R
     ↳DPs
       →DP Problem 1
         ↳Inst
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:

Termination of R could not be shown.
Duration:
0:00 minutes