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

Innermost 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
         ↳Polynomial 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


Strategy:

innermost




The following dependency pair can be strictly oriented:

IF(false, X, Y) -> ACTIVATE(Y)


There are no usable rules for innermost 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(c)=  0  
  POL(false)=  1  
  POL(true)=  0  
  POL(ACTIVATE(x1))=  x1  
  POL(F(x1))=  x1  
  POL(IF(x1, x2, x3))=  x1 + x3  

resulting in one new DP problem. 



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


Dependency Pairs:

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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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