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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a) -> H(a)
H(g(x)) -> H(f(x))
H(g(x)) -> F(x)

Furthermore, R contains one SCC. 


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


Dependency Pair:

H(g(x)) -> H(f(x))


Rules:


f(a) -> g(h(a))
h(g(x)) -> g(h(f(x)))
k(x, h(x), a) -> h(x)
k(f(x), y, x) -> f(x)


Strategy:

innermost




As we are in the innermost case, we can delete all 3 non-usable-rules.


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


Dependency Pair:

H(g(x)) -> H(f(x))


Rule:


f(a) -> g(h(a))


Strategy:

innermost




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

H(g(x)) -> H(f(x))
one new Dependency Pair is created:

H(g(a)) -> H(g(h(a)))

The transformation is resulting in no new DP problems.


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