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

A(f, a(f, a(g, a(g, x)))) -> A(g, a(g, a(g, a(f, a(f, a(f, x))))))
A(f, a(f, a(g, a(g, x)))) -> A(g, a(g, a(f, a(f, a(f, x)))))
A(f, a(f, a(g, a(g, x)))) -> A(g, a(f, a(f, a(f, x))))
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, a(f, x)))
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, x))
A(f, a(f, a(g, a(g, x)))) -> A(f, x)

Furthermore, R contains one SCC.


   R
OC
       →TRS2
DPs
           →DP Problem 1
A-Transformation


Dependency Pairs:

A(f, a(f, a(g, a(g, x)))) -> A(f, x)
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, x))
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, a(f, x)))


Rule:


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


Strategy:

innermost




We have an applicative DP problem with proper arity. Thus we can use the A-Transformation to obtain one new DP problem which consists of the A-transformed TRSs.


   R
OC
       →TRS2
DPs
           →DP Problem 1
ATrans
             ...
               →DP Problem 2
RFC Match Bounds


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




Using RFC Match Bounds, the DP problem could be solved. The Match Bound was 5.
The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Node 18 is start node and node 19 is final node.

Those nodes are connect through the following edges:



Termination of R successfully shown.
Duration:
0:01 minutes