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

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x), s(0)) -> F(g(x), g(x))
G(s(x)) -> G(x)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Polynomial Ordering


Dependency Pair:

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


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(G(x1))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 2
Dependency Graph


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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