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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


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





The following dependency pair can be strictly oriented:

ACTIVATE(nf(X)) -> F(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))=  x1  
  POL(c)=  0  
  POL(false)=  1  
  POL(true)=  0  
  POL(ACTIVATE(x1))=  1 + x1  
  POL(F(x1))=  x1  
  POL(IF(x1, x2, x3))=  x1 + x2 + x3  

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


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


Dependency Pairs:

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


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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