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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(X), X) -> F(X, a(X))
F(X, c(X)) -> F(s(X), X)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo


Dependency Pair:

F(s(X), X) -> F(X, a(X))


Rules:


f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)





The following dependency pair can be strictly oriented:

F(s(X), X) -> F(X, a(X))


Additionally, the following rules can be oriented:

f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)


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

resulting in one new DP problem. 



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


Dependency Pair:


Rules:


f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering


Dependency Pair:

F(X, c(X)) -> F(s(X), X)


Rules:


f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)





The following dependency pair can be strictly oriented:

F(X, c(X)) -> F(s(X), X)


Additionally, the following rules can be oriented:

f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 4
             ↳Dependency Graph


Dependency Pair:


Rules:


f(s(X), X) -> f(X, a(X))
f(X, c(X)) -> f(s(X), X)
f(X, X) -> c(X)





Using the Dependency Graph resulted in no new DP problems.

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