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

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(0, 1, X) -> F(g(X, X), X, X)
F(0, 1, X) -> G(X, X)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Usable Rules (Innermost)


Dependency Pair:

F(0, 1, X) -> F(g(X, X), X, X)


Rules:


f(0, 1, X) -> f(g(X, X), X, X)
g(X, Y) -> X
g(X, Y) -> Y


Strategy:

innermost




As we are in the innermost case, we can delete all 1 non-usable-rules.


   R
DPs
       →DP Problem 1
UsableRules
           →DP Problem 2
Narrowing Transformation


Dependency Pair:

F(0, 1, X) -> F(g(X, X), X, X)


Rules:


g(X, Y) -> Y
g(X, Y) -> X


Strategy:

innermost




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

F(0, 1, X) -> F(g(X, X), X, X)
one new Dependency Pair is created:

F(0, 1, X'') -> F(X'', X'', X'')

The transformation is resulting in no new DP problems.


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