Term Rewriting System R:
[X]
f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(X)) -> F(f(X))
F(g(X)) -> F(X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Polynomial Ordering


Dependency Pairs:

F(g(X)) -> F(X)
F(g(X)) -> F(f(X))


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

F(g(X)) -> F(X)
F(g(X)) -> F(f(X))


Additionally, the following usable rules for innermost can be oriented:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  1 + x1  
  POL(h(x1))=  0  
  POL(f(x1))=  x1  
  POL(F(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)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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