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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


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


Dependency Pair:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


Additionally, the following rules can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(c(x1))=  0  
  POL(s(x1))=  1 + x1  
  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(x, c(y)) -> f(x, s(f(y, y)))
f(s(x), s(y)) -> f(x, s(c(s(y))))





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(y)) -> F(y, y)


Rules:


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





The following dependency pair can be strictly oriented:

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


Additionally, the following rules can be oriented:

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


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





Using the Dependency Graph resulted in no new DP problems.

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