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
             ↳Polynomial 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)


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

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  1 + x1  
  POL(h(x1))=  0  
  POL(f(x1))=  x1  
  POL(F(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 3
             ↳Polo
             ...
               →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
                 ↳Polynomial Ordering


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))





The following dependency pair can be strictly oriented:

F'(s(s(x'''''')), s(x'''0), s(x'''0)) -> F'(s(x'''0), s(x''''''), s(s(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(F'(x1, x2, x3))=  1 + x1 + x2  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Inst
           →DP Problem 5
             ↳FwdInst
             ...
               →DP Problem 7
                 ↳Dependency Graph


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.

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