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
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS


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))


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: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
F > a
f > c

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


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


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
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and 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)


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: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
f > c > F
f > c > s

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


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