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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(cons(x, k), l) -> G(k, l, cons(x, k))
G(a, b, c) -> F(a, cons(b, c))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pairs:

G(a, b, c) -> F(a, cons(b, c))
F(cons(x, k), l) -> G(k, l, cons(x, k))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

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

F(cons(x', k'), cons(b'', c'')) -> G(k', cons(b'', c''), cons(x', k'))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(cons(x', k'), cons(b'', c'')) -> G(k', cons(b'', c''), cons(x', k'))
G(a, b, c) -> F(a, cons(b, c))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

G(a, b, c) -> F(a, cons(b, c))
one new Dependency Pair is created:

G(a', cons(b'''', c''''), cons(x''', k'''')) -> F(a', cons(cons(b'''', c''''), cons(x''', k'''')))

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 Pairs:

G(a', cons(b'''', c''''), cons(x''', k'''')) -> F(a', cons(cons(b'''', c''''), cons(x''', k'''')))
F(cons(x', k'), cons(b'', c'')) -> G(k', cons(b'', c''), cons(x', k'))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

F(cons(x', k'), cons(b'', c'')) -> G(k', cons(b'', c''), cons(x', k'))
one new Dependency Pair is created:

F(cons(x'', k''), cons(cons(b'''''', c''''''), cons(x''''', k''''''))) -> G(k'', cons(cons(b'''''', c''''''), cons(x''''', k'''''')), cons(x'', k''))

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 Pairs:

F(cons(x'', k''), cons(cons(b'''''', c''''''), cons(x''''', k''''''))) -> G(k'', cons(cons(b'''''', c''''''), cons(x''''', k'''''')), cons(x'', k''))
G(a', cons(b'''', c''''), cons(x''', k'''')) -> F(a', cons(cons(b'''', c''''), cons(x''', k'''')))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

G(a', cons(b'''', c''''), cons(x''', k'''')) -> F(a', cons(cons(b'''', c''''), cons(x''', k'''')))
one new Dependency Pair is created:

G(a'', cons(cons(b'''''''', c''''''''), cons(x''''''', k'''''''')), cons(x''''', k''''')) -> F(a'', cons(cons(cons(b'''''''', c''''''''), cons(x''''''', k'''''''')), cons(x''''', k''''')))

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 Pairs:

G(a'', cons(cons(b'''''''', c''''''''), cons(x''''''', k'''''''')), cons(x''''', k''''')) -> F(a'', cons(cons(cons(b'''''''', c''''''''), cons(x''''''', k'''''''')), cons(x''''', k''''')))
F(cons(x'', k''), cons(cons(b'''''', c''''''), cons(x''''', k''''''))) -> G(k'', cons(cons(b'''''', c''''''), cons(x''''', k'''''')), cons(x'', k''))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

F(cons(x'', k''), cons(cons(b'''''', c''''''), cons(x''''', k''''''))) -> G(k'', cons(cons(b'''''', c''''''), cons(x''''', k'''''')), cons(x'', k''))
one new Dependency Pair is created:

F(cons(x''', k'''), cons(cons(cons(b'''''''''', c''''''''''), cons(x''''''''', k'''''''''')), cons(x''''''', k''''''''))) -> G(k''', cons(cons(cons(b'''''''''', c''''''''''), cons(x''''''''', k'''''''''')), cons(x''''''', k'''''''')), cons(x''', k'''))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(cons(x''', k'''), cons(cons(cons(b'''''''''', c''''''''''), cons(x''''''''', k'''''''''')), cons(x''''''', k''''''''))) -> G(k''', cons(cons(cons(b'''''''''', c''''''''''), cons(x''''''''', k'''''''''')), cons(x''''''', k'''''''')), cons(x''', k'''))
G(a'', cons(cons(b'''''''', c''''''''), cons(x''''''', k'''''''')), cons(x''''', k''''')) -> F(a'', cons(cons(cons(b'''''''', c''''''''), cons(x''''''', k'''''''')), cons(x''''', k''''')))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

G(a'', cons(cons(b'''''''', c''''''''), cons(x''''''', k'''''''')), cons(x''''', k''''')) -> F(a'', cons(cons(cons(b'''''''', c''''''''), cons(x''''''', k'''''''')), cons(x''''', k''''')))
one new Dependency Pair is created:

G(a''', cons(cons(cons(b'''''''''''', c''''''''''''), cons(x''''''''''', k'''''''''''')), cons(x''''''''', k'''''''''')), cons(x'''''', k'''''')) -> F(a''', cons(cons(cons(cons(b'''''''''''', c''''''''''''), cons(x''''''''''', k'''''''''''')), cons(x''''''''', k'''''''''')), cons(x'''''', k'''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 7
                 ↳Instantiation Transformation


Dependency Pairs:

G(a''', cons(cons(cons(b'''''''''''', c''''''''''''), cons(x''''''''''', k'''''''''''')), cons(x''''''''', k'''''''''')), cons(x'''''', k'''''')) -> F(a''', cons(cons(cons(cons(b'''''''''''', c''''''''''''), cons(x''''''''''', k'''''''''''')), cons(x''''''''', k'''''''''')), cons(x'''''', k'''''')))
F(cons(x''', k'''), cons(cons(cons(b'''''''''', c''''''''''), cons(x''''''''', k'''''''''')), cons(x''''''', k''''''''))) -> G(k''', cons(cons(cons(b'''''''''', c''''''''''), cons(x''''''''', k'''''''''')), cons(x''''''', k'''''''')), cons(x''', k'''))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

F(cons(x''', k'''), cons(cons(cons(b'''''''''', c''''''''''), cons(x''''''''', k'''''''''')), cons(x''''''', k''''''''))) -> G(k''', cons(cons(cons(b'''''''''', c''''''''''), cons(x''''''''', k'''''''''')), cons(x''''''', k'''''''')), cons(x''', k'''))
one new Dependency Pair is created:

F(cons(x'''', k''''), cons(cons(cons(cons(b'''''''''''''', c''''''''''''''), cons(x''''''''''''', k'''''''''''''')), cons(x''''''''''', k'''''''''''')), cons(x'''''''0, k'''''''''))) -> G(k'''', cons(cons(cons(cons(b'''''''''''''', c''''''''''''''), cons(x''''''''''''', k'''''''''''''')), cons(x''''''''''', k'''''''''''')), cons(x'''''''0, k''''''''')), cons(x'''', k''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 8
                 ↳Instantiation Transformation


Dependency Pairs:

F(cons(x'''', k''''), cons(cons(cons(cons(b'''''''''''''', c''''''''''''''), cons(x''''''''''''', k'''''''''''''')), cons(x''''''''''', k'''''''''''')), cons(x'''''''0, k'''''''''))) -> G(k'''', cons(cons(cons(cons(b'''''''''''''', c''''''''''''''), cons(x''''''''''''', k'''''''''''''')), cons(x''''''''''', k'''''''''''')), cons(x'''''''0, k''''''''')), cons(x'''', k''''))
G(a''', cons(cons(cons(b'''''''''''', c''''''''''''), cons(x''''''''''', k'''''''''''')), cons(x''''''''', k'''''''''')), cons(x'''''', k'''''')) -> F(a''', cons(cons(cons(cons(b'''''''''''', c''''''''''''), cons(x''''''''''', k'''''''''''')), cons(x''''''''', k'''''''''')), cons(x'''''', k'''''')))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

G(a''', cons(cons(cons(b'''''''''''', c''''''''''''), cons(x''''''''''', k'''''''''''')), cons(x''''''''', k'''''''''')), cons(x'''''', k'''''')) -> F(a''', cons(cons(cons(cons(b'''''''''''', c''''''''''''), cons(x''''''''''', k'''''''''''')), cons(x''''''''', k'''''''''')), cons(x'''''', k'''''')))
one new Dependency Pair is created:

G(a'''', cons(cons(cons(cons(b'''''''''''''''', c''''''''''''''''), cons(x''''''''''''''', k'''''''''''''''')), cons(x''''''''''''', k'''''''''''''')), cons(x'''''''''', k''''''''''0)), cons(x''''''', k''''''')) -> F(a'''', cons(cons(cons(cons(cons(b'''''''''''''''', c''''''''''''''''), cons(x''''''''''''''', k'''''''''''''''')), cons(x''''''''''''', k'''''''''''''')), cons(x'''''''''', k''''''''''0)), cons(x''''''', k''''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 9
                 ↳Instantiation Transformation


Dependency Pairs:

G(a'''', cons(cons(cons(cons(b'''''''''''''''', c''''''''''''''''), cons(x''''''''''''''', k'''''''''''''''')), cons(x''''''''''''', k'''''''''''''')), cons(x'''''''''', k''''''''''0)), cons(x''''''', k''''''')) -> F(a'''', cons(cons(cons(cons(cons(b'''''''''''''''', c''''''''''''''''), cons(x''''''''''''''', k'''''''''''''''')), cons(x''''''''''''', k'''''''''''''')), cons(x'''''''''', k''''''''''0)), cons(x''''''', k''''''')))
F(cons(x'''', k''''), cons(cons(cons(cons(b'''''''''''''', c''''''''''''''), cons(x''''''''''''', k'''''''''''''')), cons(x''''''''''', k'''''''''''')), cons(x'''''''0, k'''''''''))) -> G(k'''', cons(cons(cons(cons(b'''''''''''''', c''''''''''''''), cons(x''''''''''''', k'''''''''''''')), cons(x''''''''''', k'''''''''''')), cons(x'''''''0, k''''''''')), cons(x'''', k''''))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

F(cons(x'''', k''''), cons(cons(cons(cons(b'''''''''''''', c''''''''''''''), cons(x''''''''''''', k'''''''''''''')), cons(x''''''''''', k'''''''''''')), cons(x'''''''0, k'''''''''))) -> G(k'''', cons(cons(cons(cons(b'''''''''''''', c''''''''''''''), cons(x''''''''''''', k'''''''''''''')), cons(x''''''''''', k'''''''''''')), cons(x'''''''0, k''''''''')), cons(x'''', k''''))
one new Dependency Pair is created:

F(cons(x''''', k'''''), cons(cons(cons(cons(cons(b'''''''''''''''''', c''''''''''''''''''), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x''''''''''''''', k'''''''''''''''')), cons(x'''''''''''0, k''''''''''''')), cons(x'''''''0', k''''''''''))) -> G(k''''', cons(cons(cons(cons(cons(b'''''''''''''''''', c''''''''''''''''''), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x''''''''''''''', k'''''''''''''''')), cons(x'''''''''''0, k''''''''''''')), cons(x'''''''0', k'''''''''')), cons(x''''', k'''''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 10
                 ↳Instantiation Transformation


Dependency Pairs:

F(cons(x''''', k'''''), cons(cons(cons(cons(cons(b'''''''''''''''''', c''''''''''''''''''), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x''''''''''''''', k'''''''''''''''')), cons(x'''''''''''0, k''''''''''''')), cons(x'''''''0', k''''''''''))) -> G(k''''', cons(cons(cons(cons(cons(b'''''''''''''''''', c''''''''''''''''''), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x''''''''''''''', k'''''''''''''''')), cons(x'''''''''''0, k''''''''''''')), cons(x'''''''0', k'''''''''')), cons(x''''', k'''''))
G(a'''', cons(cons(cons(cons(b'''''''''''''''', c''''''''''''''''), cons(x''''''''''''''', k'''''''''''''''')), cons(x''''''''''''', k'''''''''''''')), cons(x'''''''''', k''''''''''0)), cons(x''''''', k''''''')) -> F(a'''', cons(cons(cons(cons(cons(b'''''''''''''''', c''''''''''''''''), cons(x''''''''''''''', k'''''''''''''''')), cons(x''''''''''''', k'''''''''''''')), cons(x'''''''''', k''''''''''0)), cons(x''''''', k''''''')))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))





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

G(a'''', cons(cons(cons(cons(b'''''''''''''''', c''''''''''''''''), cons(x''''''''''''''', k'''''''''''''''')), cons(x''''''''''''', k'''''''''''''')), cons(x'''''''''', k''''''''''0)), cons(x''''''', k''''''')) -> F(a'''', cons(cons(cons(cons(cons(b'''''''''''''''', c''''''''''''''''), cons(x''''''''''''''', k'''''''''''''''')), cons(x''''''''''''', k'''''''''''''')), cons(x'''''''''', k''''''''''0)), cons(x''''''', k''''''')))
one new Dependency Pair is created:

G(a''''', cons(cons(cons(cons(cons(b'''''''''''''''''''', c''''''''''''''''''''), cons(x''''''''''''''''''', k'''''''''''''''''''')), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x'''''''''''''', k''''''''''''''0)), cons(x''''''''''', k''''''''''0')), cons(x'''''''', k'''''''')) -> F(a''''', cons(cons(cons(cons(cons(cons(b'''''''''''''''''''', c''''''''''''''''''''), cons(x''''''''''''''''''', k'''''''''''''''''''')), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x'''''''''''''', k''''''''''''''0)), cons(x''''''''''', k''''''''''0')), cons(x'''''''', k'''''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Inst
             ...
               →DP Problem 11
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

G(a''''', cons(cons(cons(cons(cons(b'''''''''''''''''''', c''''''''''''''''''''), cons(x''''''''''''''''''', k'''''''''''''''''''')), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x'''''''''''''', k''''''''''''''0)), cons(x''''''''''', k''''''''''0')), cons(x'''''''', k'''''''')) -> F(a''''', cons(cons(cons(cons(cons(cons(b'''''''''''''''''''', c''''''''''''''''''''), cons(x''''''''''''''''''', k'''''''''''''''''''')), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x'''''''''''''', k''''''''''''''0)), cons(x''''''''''', k''''''''''0')), cons(x'''''''', k'''''''')))
F(cons(x''''', k'''''), cons(cons(cons(cons(cons(b'''''''''''''''''', c''''''''''''''''''), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x''''''''''''''', k'''''''''''''''')), cons(x'''''''''''0, k''''''''''''')), cons(x'''''''0', k''''''''''))) -> G(k''''', cons(cons(cons(cons(cons(b'''''''''''''''''', c''''''''''''''''''), cons(x''''''''''''''''', k'''''''''''''''''')), cons(x''''''''''''''', k'''''''''''''''')), cons(x'''''''''''0, k''''''''''''')), cons(x'''''''0', k'''''''''')), cons(x''''', k'''''))


Rules:


f(empty, l) -> l
f(cons(x, k), l) -> g(k, l, cons(x, k))
g(a, b, c) -> f(a, cons(b, c))




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