Term Rewriting System R:
[X]
a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

A -> G(c)
F(g(X), b) -> F(a, X)
F(g(X), b) -> A

Furthermore, R contains one SCC. 


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


Dependency Pair:

F(g(X), b) -> F(a, X)


Rules:


a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)





The following dependency pair can be strictly oriented:

F(g(X), b) -> F(a, X)


The following rules can be oriented:

a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)


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

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
g(x1) -> g(x1)
a -> a
f(x1, x2) -> f(x1, x2)


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


Dependency Pair:


Rules:


a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)





Using the Dependency Graph resulted in no new DP problems.

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