Term Rewriting System R:
[x, y, z]
cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

INF(x) -> CONS(x, inf(s(x)))
INF(x) -> INF(s(x))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pair:

INF(x) -> INF(s(x))


Rules:


cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))


Strategy:

innermost




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

INF(x) -> INF(s(x))
one new Dependency Pair is created:

INF(s(x'')) -> INF(s(s(x'')))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

INF(s(x'')) -> INF(s(s(x'')))


Rules:


cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))


Strategy:

innermost




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

INF(s(x'')) -> INF(s(s(x'')))
one new Dependency Pair is created:

INF(s(s(x''''))) -> INF(s(s(s(x''''))))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

INF(s(s(x''''))) -> INF(s(s(s(x''''))))


Rules:


cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))


Strategy:

innermost




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

INF(s(s(x''''))) -> INF(s(s(s(x''''))))
one new Dependency Pair is created:

INF(s(s(s(x'''''')))) -> INF(s(s(s(s(x'''''')))))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

INF(s(s(s(x'''''')))) -> INF(s(s(s(s(x'''''')))))


Rules:


cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))


Strategy:

innermost




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

INF(s(s(s(x'''''')))) -> INF(s(s(s(s(x'''''')))))
one new Dependency Pair is created:

INF(s(s(s(s(x''''''''))))) -> INF(s(s(s(s(s(x''''''''))))))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

INF(s(s(s(s(x''''''''))))) -> INF(s(s(s(s(s(x''''''''))))))


Rules:


cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))


Strategy:

innermost




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

INF(s(s(s(s(x''''''''))))) -> INF(s(s(s(s(s(x''''''''))))))
one new Dependency Pair is created:

INF(s(s(s(s(s(x'''''''''')))))) -> INF(s(s(s(s(s(s(x'''''''''')))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 6
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

INF(s(s(s(s(s(x'''''''''')))))) -> INF(s(s(s(s(s(s(x'''''''''')))))))


Rules:


cons(x, cons(y, z)) -> big
inf(x) -> cons(x, inf(s(x)))


Strategy:

innermost



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