Term Rewriting System R:
[x]
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x)) -> F(x)
G(s(0)) -> G(f(s(0)))
G(s(0)) -> F(s(0))

Furthermore, R contains two SCCs. 


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


Dependency Pair:

F(s(x)) -> F(x)


Rules:


f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))





The following dependency pair can be strictly oriented:

F(s(x)) -> F(x)


Additionally, the following rules can be oriented:

f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  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 3
             ↳Dependency Graph
       →DP Problem 2
         ↳Polo


Dependency Pair:


Rules:


f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

G(s(0)) -> G(f(s(0)))


Rules:


f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))





The following dependency pair can be strictly oriented:

G(s(0)) -> G(f(s(0)))


Additionally, the following rules can be oriented:

f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(g(x1))=  0  
  POL(G(x1))=  x1  
  POL(s(x1))=  1  
  POL(f(x1))=  0  

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(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))





Using the Dependency Graph resulted in no new DP problems.

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