Term Rewriting System R:
[X, Y, Z]
f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(s(X)) -> G(X)
SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(ng(X)) -> G(activate(X))
ACTIVATE(ng(X)) -> ACTIVATE(X)

Furthermore, R contains three SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar


Dependency Pair:

G(s(X)) -> G(X)


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





The following dependency pair can be strictly oriented:

G(s(X)) -> G(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(G(x1))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 4
             ↳Dependency Graph
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar


Dependency Pair:


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polynomial Ordering
       →DP Problem 3
         ↳Nar


Dependency Pairs:

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


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





The following dependency pair can be strictly oriented:

ACTIVATE(ng(X)) -> ACTIVATE(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(n__f(x1))=  x1  
  POL(n__g(x1))=  1 + x1  
  POL(ACTIVATE(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 5
             ↳Polynomial Ordering
       →DP Problem 3
         ↳Nar


Dependency Pair:

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


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





The following dependency pair can be strictly oriented:

ACTIVATE(nf(X)) -> ACTIVATE(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(n__f(x1))=  1 + x1  
  POL(ACTIVATE(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 5
             ↳Polo
             ...
               →DP Problem 6
                 ↳Dependency Graph
       →DP Problem 3
         ↳Nar


Dependency Pair:


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





Using the Dependency Graph resulted in no new DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Narrowing Transformation


Dependency Pair:

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





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

SEL(s(X), cons(Y, Z)) -> SEL(X, activate(Z))
three new Dependency Pairs are created:

SEL(s(X), cons(Y, nf(X''))) -> SEL(X, f(activate(X'')))
SEL(s(X), cons(Y, ng(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 7
             ↳Narrowing Transformation


Dependency Pairs:

SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, ng(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, nf(X''))) -> SEL(X, f(activate(X'')))


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





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

SEL(s(X), cons(Y, nf(X''))) -> SEL(X, f(activate(X'')))
five new Dependency Pairs are created:

SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, nf(activate(X''')))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 7
             ↳Nar
             ...
               →DP Problem 8
                 ↳Narrowing Transformation


Dependency Pairs:

SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(X'''))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, ng(X''))) -> SEL(X, g(activate(X'')))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





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

SEL(s(X), cons(Y, ng(X''))) -> SEL(X, g(activate(X'')))
four new Dependency Pairs are created:

SEL(s(X), cons(Y, ng(X'''))) -> SEL(X, ng(activate(X''')))
SEL(s(X), cons(Y, ng(nf(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(X), cons(Y, ng(ng(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, ng(X'''))) -> SEL(X, g(X'''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 7
             ↳Nar
             ...
               →DP Problem 9
                 ↳Polynomial Ordering


Dependency Pairs:

SEL(s(X), cons(Y, ng(X'''))) -> SEL(X, g(X'''))
SEL(s(X), cons(Y, ng(ng(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, ng(nf(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(X'''))


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





The following dependency pairs can be strictly oriented:

SEL(s(X), cons(Y, ng(X'''))) -> SEL(X, g(X'''))
SEL(s(X), cons(Y, ng(ng(X''')))) -> SEL(X, g(g(activate(X'''))))
SEL(s(X), cons(Y, ng(nf(X''')))) -> SEL(X, g(f(activate(X'''))))
SEL(s(X), cons(Y, nf(nf(X''')))) -> SEL(X, f(f(activate(X'''))))
SEL(s(X), cons(Y, nf(ng(X''')))) -> SEL(X, f(g(activate(X'''))))
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, cons(activate(X'''), nf(ng(activate(X''')))))
SEL(s(X), cons(Y, Z')) -> SEL(X, Z')
SEL(s(X), cons(Y, nf(X'''))) -> SEL(X, f(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(activate(x1))=  0  
  POL(n__f(x1))=  0  
  POL(0)=  0  
  POL(g(x1))=  0  
  POL(cons(x1, x2))=  0  
  POL(SEL(x1, x2))=  x1  
  POL(s(x1))=  1 + x1  
  POL(n__g(x1))=  0  
  POL(f(x1))=  0  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
       →DP Problem 3
         ↳Nar
           →DP Problem 7
             ↳Nar
             ...
               →DP Problem 10
                 ↳Dependency Graph


Dependency Pair:


Rules:


f(X) -> cons(X, nf(ng(X)))
f(X) -> nf(X)
g(0) -> s(0)
g(s(X)) -> s(s(g(X)))
g(X) -> ng(X)
sel(0, cons(X, Y)) -> X
sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
activate(nf(X)) -> f(activate(X))
activate(ng(X)) -> g(activate(X))
activate(X) -> X





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:01 minutes