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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Size-Change Principle


Dependency Pairs:

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


Rules:


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





We number the DPs as follows:
  1. A(h, x) -> A(f, x)
  2. A(f, a(f, x)) -> A(x, x)
and get the following Size-Change Graph(s):
{1} , {1}
2=2
{2} , {2}
2>1
2>2

which lead(s) to this/these maximal multigraph(s):
{2} , {2}
2>1
2>2
{1} , {2}
2>1
2>2
{2} , {1}
2>2

DP: empty set
Oriented Rules: none

We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial

with Argument Filtering System:
a(x1, x2) -> a(x1, x2)

We obtain no new DP problems.

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