Term Rewriting System R:
[X]
f(X) -> g(nh(nf(X)))
f(X) -> nf(X)
h(X) -> nh(X)
activate(nh(X)) -> h(activate(X))
activate(nf(X)) -> f(activate(X))
activate(X) -> X

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

ACTIVATE(nh(X)) -> H(activate(X))
ACTIVATE(nh(X)) -> ACTIVATE(X)
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(nh(X)) -> ACTIVATE(X)


Rules:


f(X) -> g(nh(nf(X)))
f(X) -> nf(X)
h(X) -> nh(X)
activate(nh(X)) -> h(activate(X))
activate(nf(X)) -> f(activate(X))
activate(X) -> X





The following dependency pair can be strictly oriented:

ACTIVATE(nh(X)) -> ACTIVATE(X)


The following rules can be oriented:

f(X) -> g(nh(nf(X)))
f(X) -> nf(X)
h(X) -> nh(X)
activate(nh(X)) -> h(activate(X))
activate(nf(X)) -> f(activate(X))
activate(X) -> X


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__h(x1))=  1 + x1  
  POL(n__f(x1))=  x1  
  POL(activate(x1))=  x1  
  POL(g)=  0  
  POL(h(x1))=  1 + x1  
  POL(ACTIVATE(x1))=  x1  
  POL(f(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
nh(x1) -> nh(x1)
nf(x1) -> nf(x1)
f(x1) -> f(x1)
g(x1) -> g
h(x1) -> h(x1)
activate(x1) -> activate(x1)


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


Dependency Pair:

ACTIVATE(nf(X)) -> ACTIVATE(X)


Rules:


f(X) -> g(nh(nf(X)))
f(X) -> nf(X)
h(X) -> nh(X)
activate(nh(X)) -> h(activate(X))
activate(nf(X)) -> f(activate(X))
activate(X) -> X





The following dependency pair can be strictly oriented:

ACTIVATE(nf(X)) -> ACTIVATE(X)


The following rules can be oriented:

f(X) -> g(nh(nf(X)))
f(X) -> nf(X)
h(X) -> nh(X)
activate(nh(X)) -> h(activate(X))
activate(nf(X)) -> f(activate(X))
activate(X) -> X


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__h(x1))=  x1  
  POL(n__f(x1))=  1 + x1  
  POL(activate(x1))=  x1  
  POL(g(x1))=  x1  
  POL(h(x1))=  x1  
  POL(ACTIVATE(x1))=  1 + x1  
  POL(f(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
nf(x1) -> nf(x1)
f(x1) -> f(x1)
g(x1) -> g(x1)
nh(x1) -> nh(x1)
h(x1) -> h(x1)
activate(x1) -> activate(x1)


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


Dependency Pair:


Rules:


f(X) -> g(nh(nf(X)))
f(X) -> nf(X)
h(X) -> nh(X)
activate(nh(X)) -> h(activate(X))
activate(nf(X)) -> f(activate(X))
activate(X) -> X





Using the Dependency Graph resulted in no new DP problems.

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