Term Rewriting System R:
[x]
f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Polynomial Ordering


Dependency Pair:

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


Rules:


f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))





The following dependency pair can be strictly oriented:

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


Additionally, the following rules can be oriented:

f(0) -> 1
f(s(x)) -> g(f(x))
f(s(x)) -> +(f(x), s(f(x)))
g(x) -> +(x, s(x))


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





Using the Dependency Graph resulted in no new DP problems.

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