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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(g(x)) -> G(h(g(x)))
G(g(x)) -> H(g(x))
H(h(x)) -> H(f(h(x), x))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pair:

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


Rules:


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





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

G(g(x)) -> G(h(g(x)))
two new Dependency Pairs are created:

G(g(h(g(x'')))) -> G(h(g(x'')))
G(g(g(x''))) -> G(h(g(h(g(x'')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

G(g(g(x''))) -> G(h(g(h(g(x'')))))
G(g(h(g(x'')))) -> G(h(g(x'')))


Rules:


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





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

G(g(h(g(x'')))) -> G(h(g(x'')))
two new Dependency Pairs are created:

G(g(h(g(h(g(x')))))) -> G(h(g(x')))
G(g(h(g(g(x'))))) -> G(h(g(h(g(x')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

G(g(h(g(g(x'))))) -> G(h(g(h(g(x')))))
G(g(h(g(h(g(x')))))) -> G(h(g(x')))
G(g(g(x''))) -> G(h(g(h(g(x'')))))


Rules:


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





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

G(g(g(x''))) -> G(h(g(h(g(x'')))))
three new Dependency Pairs are created:

G(g(g(x'''))) -> G(h(g(x''')))
G(g(g(h(g(x'))))) -> G(h(g(h(g(x')))))
G(g(g(g(x')))) -> G(h(g(h(g(h(g(x')))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Argument Filtering and Ordering


Dependency Pairs:

G(g(g(g(x')))) -> G(h(g(h(g(h(g(x')))))))
G(g(g(h(g(x'))))) -> G(h(g(h(g(x')))))
G(g(g(x'''))) -> G(h(g(x''')))
G(g(h(g(h(g(x')))))) -> G(h(g(x')))
G(g(h(g(g(x'))))) -> G(h(g(h(g(x')))))


Rules:


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





The following dependency pairs can be strictly oriented:

G(g(g(h(g(x'))))) -> G(h(g(h(g(x')))))
G(g(g(x'''))) -> G(h(g(x''')))
G(g(h(g(h(g(x')))))) -> G(h(g(x')))
G(g(h(g(g(x'))))) -> G(h(g(h(g(x')))))


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

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Argument Filtering and Ordering


Dependency Pair:

G(g(g(g(x')))) -> G(h(g(h(g(h(g(x')))))))


Rules:


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





The following dependency pair can be strictly oriented:

G(g(g(g(x')))) -> G(h(g(h(g(h(g(x')))))))


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

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
g > h

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


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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