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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pair:

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


Rules:


f(s(x)) -> s(s(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
             ↳Polynomial Ordering


Dependency Pair:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(F(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes