Term Rewriting System R:
[a, x, k, d]
f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a, empty) -> G(a, empty)
F(a, cons(x, k)) -> F(cons(x, a), k)
G(cons(x, k), d) -> G(k, cons(x, d))

Furthermore, R contains two SCCs. 


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


Dependency Pair:

G(cons(x, k), d) -> G(k, cons(x, d))


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))





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

G(cons(x, k), d) -> G(k, cons(x, d))
one new Dependency Pair is created:

G(cons(x0, k''), cons(x'', d'')) -> G(k'', cons(x0, cons(x'', d'')))

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(cons(x0, k''), cons(x'', d'')) -> G(k'', cons(x0, cons(x'', d'')))


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))





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

G(cons(x0, k''), cons(x'', d'')) -> G(k'', cons(x0, cons(x'', d'')))
one new Dependency Pair is created:

G(cons(x0'', k''''), cons(x'''', cons(x''''', d''''))) -> G(k'''', cons(x0'', cons(x'''', cons(x''''', d''''))))

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(cons(x0'', k''''), cons(x'''', cons(x''''', d''''))) -> G(k'''', cons(x0'', cons(x'''', cons(x''''', d''''))))


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))





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

G(cons(x0'', k''''), cons(x'''', cons(x''''', d''''))) -> G(k'''', cons(x0'', cons(x'''', cons(x''''', d''''))))
one new Dependency Pair is created:

G(cons(x0'''', k''''''), cons(x''''0, cons(x'''''0, cons(x'''''''', d'''''')))) -> G(k'''''', cons(x0'''', cons(x''''0, cons(x'''''0, cons(x'''''''', d'''''')))))

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(cons(x0'''', k''''''), cons(x''''0, cons(x'''''0, cons(x'''''''', d'''''')))) -> G(k'''''', cons(x0'''', cons(x''''0, cons(x'''''0, cons(x'''''''', d'''''')))))


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))





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

G(cons(x0'''', k''''''), cons(x''''0, cons(x'''''0, cons(x'''''''', d'''''')))) -> G(k'''''', cons(x0'''', cons(x''''0, cons(x'''''0, cons(x'''''''', d'''''')))))
one new Dependency Pair is created:

G(cons(x0'''''', k''''''''), cons(x''''0'', cons(x'''''0'', cons(x'''''''''', cons(x''''''''''', d''''''''))))) -> G(k'''''''', cons(x0'''''', cons(x''''0'', cons(x'''''0'', cons(x'''''''''', cons(x''''''''''', d''''''''))))))

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(cons(x0'''''', k''''''''), cons(x''''0'', cons(x'''''0'', cons(x'''''''''', cons(x''''''''''', d''''''''))))) -> G(k'''''''', cons(x0'''''', cons(x''''0'', cons(x'''''0'', cons(x'''''''''', cons(x''''''''''', d''''''''))))))


Rules:


f(a, empty) -> g(a, empty)
f(a, cons(x, k)) -> f(cons(x, a), k)
g(empty, d) -> d
g(cons(x, k), d) -> g(k, cons(x, d))





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

G(cons(x0'''''', k''''''''), cons(x''''0'', cons(x'''''0'', cons(x'''''''''', cons(x''''''''''', d''''''''))))) -> G(k'''''''', cons(x0'''''', cons(x''''0'', cons(x'''''0'', cons(x'''''''''', cons(x''''''''''', d''''''''))))))
one new Dependency Pair is created:

G(cons(x0'''''''', k''''''''''), cons(x''''0'''', cons(x'''''0'''', cons(x''''''''''0, cons(x'''''''''''0, cons(x'''''''''''''', d'''''''''')))))) -> G(k'''''''''', cons(x0'''''''', cons(x''''0'''', cons(x'''''0'''', cons(x''''''''''0, cons(x'''''''''''0, cons(x'''''''''''''', d'''''''''')))))))

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