Term Rewriting System R:
[X, Y]
*(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X)
*(X, 1) -> X
*(X, 0) -> X
*(X, 0) -> 0

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

*'(X, +(Y, 1)) -> *'(X, +(Y, *(1, 0)))
*'(X, +(Y, 1)) -> *'(1, 0)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pair:

*'(X, +(Y, 1)) -> *'(X, +(Y, *(1, 0)))


Rules:


*(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X)
*(X, 1) -> X
*(X, 0) -> X
*(X, 0) -> 0





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

*'(X, +(Y, 1)) -> *'(X, +(Y, *(1, 0)))
two new Dependency Pairs are created:

*'(X, +(Y, 1)) -> *'(X, +(Y, 1))
*'(X, +(Y, 1)) -> *'(X, +(Y, 0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

*'(X, +(Y, 1)) -> *'(X, +(Y, 1))


Rules:


*(X, +(Y, 1)) -> +(*(X, +(Y, *(1, 0))), X)
*(X, 1) -> X
*(X, 0) -> X
*(X, 0) -> 0




Termination of R could not be shown.
Duration:
0:00 minutes