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

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Polynomial Ordering


Dependency Pair:

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


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))





The following dependency pair can be strictly oriented:

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


Additionally, the following rules can be oriented:

f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))


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

resulting in one new DP problem.



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


Dependency Pair:


Rules:


f(c(X, s(Y))) -> f(c(s(X), Y))
g(c(s(X), Y)) -> f(c(X, s(Y)))





Using the Dependency Graph resulted in no new DP problems.

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