Term Rewriting System R:
[]
g(a) -> g(b)
b -> f(a, a)
f(a, a) -> g(d)

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

G(a) -> G(b)
G(a) -> B
B -> F(a, a)
F(a, a) -> G(d)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering


Dependency Pair:

G(a) -> G(b)


Rules:


g(a) -> g(b)
b -> f(a, a)
f(a, a) -> g(d)





The following dependency pair can be strictly oriented:

G(a) -> G(b)


The following usable rules w.r.t. to the AFS can be oriented:

g(a) -> g(b)
b -> f(a, a)
f(a, a) -> g(d)


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

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


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


Dependency Pair:


Rules:


g(a) -> g(b)
b -> f(a, a)
f(a, a) -> g(d)





Using the Dependency Graph resulted in no new DP problems.

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