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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pairs:

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


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


The following usable rules for innermost w.r.t. to the AFS can be oriented:

f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))


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

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
s(x1) -> s(x1)
f(x1) -> f(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rules:


f(x) -> s(x)
f(s(s(x))) -> s(f(f(x)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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