Term Rewriting System R:
[x]
f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(1) -> F(g(1))
F(1) -> G(1)
G(0) -> G(f(0))
G(0) -> F(0)

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar


Dependency Pair:

F(1) -> F(g(1))


Rules:


f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)


Strategy:

innermost




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

F(1) -> F(g(1))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Narrowing Transformation


Dependency Pair:

G(0) -> G(f(0))


Rules:


f(1) -> f(g(1))
f(f(x)) -> f(x)
g(0) -> g(f(0))
g(g(x)) -> g(x)


Strategy:

innermost




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

G(0) -> G(f(0))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.


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