Term Rewriting System R:
[x]
f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x)) -> F(f(p(s(x))))
F(s(x)) -> F(p(s(x)))
F(s(x)) -> P(s(x))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

F(s(x)) -> F(p(s(x)))
F(s(x)) -> F(f(p(s(x))))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x





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

F(s(x)) -> F(f(p(s(x))))
one new Dependency Pair is created:

F(s(x'')) -> F(f(x''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

F(s(x'')) -> F(f(x''))
F(s(x)) -> F(p(s(x)))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x





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

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

F(s(x'')) -> F(x'')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

F(s(x'')) -> F(x'')
F(s(x'')) -> F(f(x''))


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x





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

F(s(x'')) -> F(f(x''))
two new Dependency Pairs are created:

F(s(s(x'))) -> F(s(f(f(p(s(x'))))))
F(s(0)) -> F(0)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(s(s(x'))) -> F(s(f(f(p(s(x'))))))
F(s(x'')) -> F(x'')


Rules:


f(s(x)) -> s(f(f(p(s(x)))))
f(0) -> 0
p(s(x)) -> x




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