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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)


Dependency Pairs:

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


Rules:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 4 non-usable-rules.


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳Instantiation Transformation


Dependency Pairs:

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


Rule:

none


Strategy:

innermost




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

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

H(0, 0) -> F(0, 0, 0)

The transformation is resulting in no new DP problems.


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