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

Innermost Termination of R to be shown.



   TRS
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.


   TRS
DPs
       →DP Problem 1
Modular Removal of Rules


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




We have the following set of usable rules:

f(s(s(x))) -> s(f(f(x)))
f(x) -> s(x)
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(F(x1))=  1 + x1  
  POL(f(x1))=  1 + x1  

We have the following set D of usable symbols: {s, F, f}
The following Dependency Pairs can be deleted as the lhs is strictly greater than the corresponding rhs:

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

No Rules can be deleted.

After the removal, there are no SCCs in the dependency graph which results in no DP problems which have to be solved.


Innermost Termination of R successfully shown.
Duration:
0:01 minutes