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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(a) -> G(b)
G(a) -> B
B -> F(a, a)
F(a, a) -> G(d)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Rewriting Transformation


Dependency Pair:

G(a) -> G(b)


Rules:


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


Strategy:

innermost




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

G(a) -> G(b)
one new Dependency Pair is created:

G(a) -> G(f(a, a))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rewriting Transformation


Dependency Pair:

G(a) -> G(f(a, a))


Rules:


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


Strategy:

innermost




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

G(a) -> G(f(a, a))
one new Dependency Pair is created:

G(a) -> G(g(d))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pair:

G(a) -> G(g(d))


Rules:


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


Strategy:

innermost




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

G(a) -> G(g(d))
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