Term Rewriting System R:
[y]
f(a, y) -> f(y, g(y))
g(a) -> b
g(b) -> b

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a, y) -> F(y, g(y))
F(a, y) -> G(y)

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)


Dependency Pair:

F(a, y) -> F(y, g(y))


Rules:


f(a, y) -> f(y, g(y))
g(a) -> b
g(b) -> b


Strategy:

innermost




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


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Narrowing Transformation


Dependency Pair:

F(a, y) -> F(y, g(y))


Rules:


g(b) -> b
g(a) -> b


Strategy:

innermost




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

F(a, y) -> F(y, g(y))
two new Dependency Pairs are created:

F(a, b) -> F(b, b)
F(a, a) -> F(a, b)

The transformation is resulting in no new DP problems.


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