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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(a, x), a) -> F(a, f(f(x, f(a, a)), a))
F(f(a, x), a) -> F(f(x, f(a, a)), a)
F(f(a, x), a) -> F(x, f(a, a))
F(f(a, x), a) -> F(a, a)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

F(f(a, x), a) -> F(x, f(a, a))
F(f(a, x), a) -> F(f(x, f(a, a)), a)


Rule:


f(f(a, x), a) -> f(a, f(f(x, f(a, a)), a))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(f(a, x), a) -> F(x, f(a, a))


Additionally, the following usable rule for innermost w.r.t. to the implicit AFS can be oriented:

f(f(a, x), a) -> f(a, f(f(x, f(a, a)), a))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(a)=  1  
  POL(f(x1, x2))=  0  
  POL(F(x1, x2))=  x2  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(f(a, x), a) -> F(f(x, f(a, a)), a)


Rule:


f(f(a, x), a) -> f(a, f(f(x, f(a, a)), a))


Strategy:

innermost



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