Term Rewriting System R:
[X, Y]
f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(X) -> IF(X, c, nf(ntrue))
IF(false, X, Y) -> ACTIVATE(Y)
ACTIVATE(nf(X)) -> F(activate(X))
ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(ntrue) -> TRUE

Furthermore, R contains one SCC. 


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


Dependency Pairs:

ACTIVATE(nf(X)) -> ACTIVATE(X)
ACTIVATE(nf(X)) -> F(activate(X))
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X





The following dependency pair can be strictly oriented:

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


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

activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
true -> ntrue


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(n__f(x1))=  x1  
  POL(activate(x1))=  x1  
  POL(n__true)=  0  
  POL(c)=  0  
  POL(if(x1, x2, x3))=  x1 + x2 + x3  
  POL(false)=  1  
  POL(true)=  0  
  POL(ACTIVATE(x1))=  1 + x1  
  POL(F(x1))=  x1  
  POL(IF(x1, x2, x3))=  x1 + x2 + x3  
  POL(f(x1))=  x1  

resulting in one new DP problem.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
F(x1) -> F(x1)
nf(x1) -> nf(x1)
activate(x1) -> activate(x1)
IF(x1, x2, x3) -> IF(x1, x2, x3)
f(x1) -> f(x1)
true -> true
if(x1, x2, x3) -> if(x1, x2, x3)


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


Dependency Pairs:

ACTIVATE(nf(X)) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
F(X) -> IF(X, c, nf(ntrue))


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X





Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


Dependency Pair:

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


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
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 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.
Used Argument Filtering System:
ACTIVATE(x1) -> ACTIVATE(x1)
nf(x1) -> nf(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 4
                 ↳Dependency Graph


Dependency Pair:


Rules:


f(X) -> if(X, c, nf(ntrue))
f(X) -> nf(X)
if(true, X, Y) -> X
if(false, X, Y) -> activate(Y)
true -> ntrue
activate(nf(X)) -> f(activate(X))
activate(ntrue) -> true
activate(X) -> X





Using the Dependency Graph resulted in no new DP problems.

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