Term Rewriting System R:
[x]
f(x, x) -> f(g(x), x)
g(x) -> s(x)

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(x, x) -> F(g(x), x)
F(x, x) -> G(x)

Furthermore, R contains one SCC. 


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


Dependency Pair:

F(x, x) -> F(g(x), x)


Rules:


f(x, x) -> f(g(x), x)
g(x) -> s(x)


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
                 ↳Rewriting Transformation


Dependency Pair:

F(x, x) -> F(g(x), x)


Rule:


g(x) -> s(x)


Strategy:

innermost




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

F(x, x) -> F(g(x), x)
one new Dependency Pair is created:

F(x, x) -> F(s(x), x)

The transformation is resulting in no new DP problems.


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