Term Rewriting System R:
[x]
f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a, x) -> F(g(x), x)
F(a, x) -> G(x)
H(g(x)) -> H(a)
G(h(x)) -> G(x)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳Remaining


Dependency Pair:

G(h(x)) -> G(x)


Rules:


f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)





The following dependency pair can be strictly oriented:

G(h(x)) -> G(x)


The following rules can be oriented:

f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{a, g}

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
h(x1) -> h(x1)
f(x1, x2) -> f(x1, x2)
g(x1) -> g


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳Remaining


Dependency Pair:


Rules:


f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(a, x) -> F(g(x), x)


Rules:


f(a, x) -> f(g(x), x)
h(g(x)) -> h(a)
h(h(x)) -> x
g(h(x)) -> g(x)




Termination of R could not be shown.
Duration:
0:00 minutes