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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(h(x)) -> F(i(x))
F(h(x)) -> I(x)
G(i(x)) -> G(h(x))
G(i(x)) -> H(x)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

F(h(x)) -> F(i(x))


Rules:


f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b





The following dependency pair can be strictly oriented:

F(h(x)) -> F(i(x))


The following usable rule using the Ce-refinement can be oriented:

i(a) -> b


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
h > {i, b}

resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
h(x1) -> h(x1)
i(x1) -> i(x1)


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


Dependency Pair:


Rules:


f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

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


Rules:


f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b





The following dependency pair can be strictly oriented:

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


The following usable rule using the Ce-refinement can be oriented:

h(a) -> b


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
i > h
{a, b}

resulting in one new DP problem.
Used Argument Filtering System:
G(x1) -> G(x1)
i(x1) -> i(x1)
h(x1) -> h(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
       →DP Problem 2
         ↳AFS
           →DP Problem 4
             ↳Dependency Graph


Dependency Pair:


Rules:


f(h(x)) -> f(i(x))
g(i(x)) -> g(h(x))
h(a) -> b
i(a) -> b





Using the Dependency Graph resulted in no new DP problems.

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