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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x)) -> F(f(x))
F(g(x)) -> F(x)
F'(s(x), y, y) -> F'(y, x, s(x))

Furthermore, R contains two SCCs. 


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


Dependency Pairs:

F(g(x)) -> F(x)
F(g(x)) -> F(f(x))


Rules:


f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))





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

F(g(x)) -> F(f(x))
two new Dependency Pairs are created:

F(g(g(x''))) -> F(g(f(f(x''))))
F(g(h(x''))) -> F(h(g(x'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 3
             ↳Argument Filtering and Ordering
       →DP Problem 2
         ↳Inst


Dependency Pairs:

F(g(g(x''))) -> F(g(f(f(x''))))
F(g(x)) -> F(x)


Rules:


f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))





The following dependency pairs can be strictly oriented:

F(g(g(x''))) -> F(g(f(f(x''))))
F(g(x)) -> F(x)


The following rules can be oriented:

f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))


Used ordering: Homeomorphic Embedding Order with EMB
resulting in one new DP problem.
Used Argument Filtering System:
F(x1) -> F(x1)
g(x1) -> g(x1)
f(x1) -> x1
h(x1) -> h
f'(x1, x2, x3) -> f'


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 3
             ↳AFS
             ...
               →DP Problem 4
                 ↳Dependency Graph
       →DP Problem 2
         ↳Inst


Dependency Pair:


Rules:


f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

F'(s(x), y, y) -> F'(y, x, s(x))


Rules:


f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))





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

F'(s(x), y, y) -> F'(y, x, s(x))
one new Dependency Pair is created:

F'(s(x0), s(x'''), s(x''')) -> F'(s(x'''), x0, s(x0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Inst
           →DP Problem 5
             ↳Forward Instantiation Transformation


Dependency Pair:

F'(s(x0), s(x'''), s(x''')) -> F'(s(x'''), x0, s(x0))


Rules:


f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))





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

F'(s(x0), s(x'''), s(x''')) -> F'(s(x'''), x0, s(x0))
one new Dependency Pair is created:

F'(s(s(x'''''')), s(x'''0), s(x'''0)) -> F'(s(x'''0), s(x''''''), s(s(x'''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Inst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Instantiation Transformation


Dependency Pair:

F'(s(s(x'''''')), s(x'''0), s(x'''0)) -> F'(s(x'''0), s(x''''''), s(s(x'''''')))


Rules:


f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))





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

F'(s(s(x'''''')), s(x'''0), s(x'''0)) -> F'(s(x'''0), s(x''''''), s(s(x'''''')))
one new Dependency Pair is created:

F'(s(s(x''''''0)), s(s(x''''''''')), s(s(x'''''''''))) -> F'(s(s(x''''''''')), s(x''''''0), s(s(x''''''0)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Inst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 7
                 ↳Forward Instantiation Transformation


Dependency Pair:

F'(s(s(x''''''0)), s(s(x''''''''')), s(s(x'''''''''))) -> F'(s(s(x''''''''')), s(x''''''0), s(s(x''''''0)))


Rules:


f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))





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

F'(s(s(x''''''0)), s(s(x''''''''')), s(s(x'''''''''))) -> F'(s(s(x''''''''')), s(x''''''0), s(s(x''''''0)))
one new Dependency Pair is created:

F'(s(s(s(x''''''''''''))), s(s(x'''''''''0)), s(s(x'''''''''0))) -> F'(s(s(x'''''''''0)), s(s(x'''''''''''')), s(s(s(x''''''''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Inst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 8
                 ↳Instantiation Transformation


Dependency Pair:

F'(s(s(s(x''''''''''''))), s(s(x'''''''''0)), s(s(x'''''''''0))) -> F'(s(s(x'''''''''0)), s(s(x'''''''''''')), s(s(s(x''''''''''''))))


Rules:


f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))





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

F'(s(s(s(x''''''''''''))), s(s(x'''''''''0)), s(s(x'''''''''0))) -> F'(s(s(x'''''''''0)), s(s(x'''''''''''')), s(s(s(x''''''''''''))))
one new Dependency Pair is created:

F'(s(s(s(x''''''''''''0))), s(s(s(x'''''''''''''''))), s(s(s(x''''''''''''''')))) -> F'(s(s(s(x'''''''''''''''))), s(s(x''''''''''''0)), s(s(s(x''''''''''''0))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Inst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 9
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F'(s(s(s(x''''''''''''0))), s(s(s(x'''''''''''''''))), s(s(s(x''''''''''''''')))) -> F'(s(s(s(x'''''''''''''''))), s(s(x''''''''''''0)), s(s(s(x''''''''''''0))))


Rules:


f(g(x)) -> g(f(f(x)))
f(h(x)) -> h(g(x))
f'(s(x), y, y) -> f'(y, x, s(x))




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