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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(c(s(x), y)) -> F(c(x, s(y)))
G(c(x, s(y))) -> G(c(s(x), y))
G(s(f(x))) -> G(f(x))

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS


Dependency Pair:

F(c(s(x), y)) -> F(c(x, s(y)))


Rules:


f(c(s(x), y)) -> f(c(x, s(y)))
g(c(x, s(y))) -> g(c(s(x), y))
g(s(f(x))) -> g(f(x))





The following dependency pair can be strictly oriented:

F(c(s(x), y)) -> F(c(x, s(y)))


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
c > s

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 4
             ↳Dependency Graph
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS


Dependency Pair:


Rules:


f(c(s(x), y)) -> f(c(x, s(y)))
g(c(x, s(y))) -> g(c(s(x), y))
g(s(f(x))) -> g(f(x))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Argument Filtering and Ordering
       →DP Problem 3
         ↳AFS


Dependency Pair:

G(c(x, s(y))) -> G(c(s(x), y))


Rules:


f(c(s(x), y)) -> f(c(x, s(y)))
g(c(x, s(y))) -> g(c(s(x), y))
g(s(f(x))) -> g(f(x))





The following dependency pair can be strictly oriented:

G(c(x, s(y))) -> G(c(s(x), y))


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 5
             ↳Dependency Graph
       →DP Problem 3
         ↳AFS


Dependency Pair:


Rules:


f(c(s(x), y)) -> f(c(x, s(y)))
g(c(x, s(y))) -> g(c(s(x), y))
g(s(f(x))) -> g(f(x))





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳Argument Filtering and Ordering


Dependency Pair:

G(s(f(x))) -> G(f(x))


Rules:


f(c(s(x), y)) -> f(c(x, s(y)))
g(c(x, s(y))) -> g(c(s(x), y))
g(s(f(x))) -> g(f(x))





The following dependency pair can be strictly oriented:

G(s(f(x))) -> G(f(x))


The following usable rule using the Ce-refinement can be oriented:

f(c(s(x), y)) -> f(c(x, s(y)))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
c > s

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
       →DP Problem 3
         ↳AFS
           →DP Problem 6
             ↳Dependency Graph


Dependency Pair:


Rules:


f(c(s(x), y)) -> f(c(x, s(y)))
g(c(x, s(y))) -> g(c(s(x), y))
g(s(f(x))) -> g(f(x))





Using the Dependency Graph resulted in no new DP problems.

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