Term Rewriting System R:
[x, y]
ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

AP(f, x) -> AP(f, app(g, x))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pair:

AP(f, x) -> AP(f, app(g, x))


Rules:


ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))


Strategy:

innermost




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

AP(f, x) -> AP(f, app(g, x))
one new Dependency Pair is created:

AP(f, app(g, x'')) -> AP(f, app(g, app(g, x'')))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

AP(f, app(g, x'')) -> AP(f, app(g, app(g, x'')))


Rules:


ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))


Strategy:

innermost




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

AP(f, app(g, x'')) -> AP(f, app(g, app(g, x'')))
one new Dependency Pair is created:

AP(f, app(g, app(g, x''''))) -> AP(f, app(g, app(g, app(g, 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:

AP(f, app(g, app(g, x''''))) -> AP(f, app(g, app(g, app(g, x''''))))


Rules:


ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))


Strategy:

innermost




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

AP(f, app(g, app(g, x''''))) -> AP(f, app(g, app(g, app(g, x''''))))
one new Dependency Pair is created:

AP(f, app(g, app(g, app(g, x'''''')))) -> AP(f, app(g, app(g, app(g, app(g, 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:

AP(f, app(g, app(g, app(g, x'''''')))) -> AP(f, app(g, app(g, app(g, app(g, x'''''')))))


Rules:


ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))


Strategy:

innermost




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

AP(f, app(g, app(g, app(g, x'''''')))) -> AP(f, app(g, app(g, app(g, app(g, x'''''')))))
one new Dependency Pair is created:

AP(f, app(g, app(g, app(g, app(g, x''''''''))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, 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:

AP(f, app(g, app(g, app(g, app(g, x''''''''))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, x''''''''))))))


Rules:


ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))


Strategy:

innermost




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

AP(f, app(g, app(g, app(g, app(g, x''''''''))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, x''''''''))))))
one new Dependency Pair is created:

AP(f, app(g, app(g, app(g, app(g, app(g, x'''''''''')))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, app(g, 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:

AP(f, app(g, app(g, app(g, app(g, app(g, x'''''''''')))))) -> AP(f, app(g, app(g, app(g, app(g, app(g, app(g, x'''''''''')))))))


Rules:


ap(ap(g, x), y) -> y
ap(f, x) -> ap(f, app(g, x))


Strategy:

innermost



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