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)

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

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

where the Polynomial interpretation:
  POL(g(x1))=  x1  
  POL(h(x1))=  x1  
  POL(a)=  0  
  POL(f(x1))=  x1  
  POL(k(x1, x2, x3))=  1 + x1 + x2 + x3  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳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
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳DPs
             ...
               →DP Problem 1
                 ↳Usable Rules (Innermost)


Dependency Pair:

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


Rules:


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


Strategy:

innermost




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


   R
     ↳RRRPolo
       →TRS2
         ↳OC
           →TRS3
             ↳DPs
             ...
               →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.


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