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

Innermost 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
Usable Rules (Innermost)


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


Strategy:

innermost




As we are in the innermost case, we can delete all 2 non-usable-rules.


   R
DPs
       →DP Problem 1
UsableRules
           →DP Problem 2
Non Termination


Dependency Pair:

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


Rules:


*(X, 0) -> 0
*(X, 0) -> X


Strategy:

innermost




Found an infinite P-chain over R:
P =

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

R =

*(X, 0) -> 0
*(X, 0) -> X

s = *'(X', +(Y', *(1, 0)))
evaluates to t =*'(X', +(Y', *(1, 0)))

Thus, s starts an infinite chain.

Innermost Non-Termination of R could be shown.
Duration:
0:03 minutes