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))

Innermost 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))


Strategy:

innermost




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

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


Rules:


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


Strategy:

innermost




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

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

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

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


Rules:


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


Strategy:

innermost




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

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

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

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


Rules:


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


Strategy:

innermost




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

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

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

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


Rules:


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


Strategy:

innermost




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

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

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:

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