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
Polynomial Ordering
       →DP Problem 2
Polo


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)


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

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(G(x1))=  x1  
  POL(h(x1))=  1 + 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(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
Polo
       →DP Problem 2
Polynomial Ordering


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)





The following dependency pair can be strictly oriented:

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


Additionally, the following usable rule w.r.t. to the implicit AFS can be oriented:

g(h(x)) -> g(x)


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  0  
  POL(h(x1))=  0  
  POL(a)=  1  
  POL(F(x1, x2))=  x1  

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(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.

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