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

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:

F(g(x), a) -> F(x, g(a))
F(g(x), g(y)) -> H(g(y), x, g(y))
H(g(x), y, z) -> F(y, h(x, y, z))
H(g(x), y, z) -> H(x, y, z)

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Size-Change Principle


Dependency Pairs:

H(g(x), y, z) -> H(x, y, z)
H(g(x), y, z) -> F(y, h(x, y, z))
F(g(x), g(y)) -> H(g(y), x, g(y))
F(g(x), a) -> F(x, g(a))


Rules:


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


Strategy:

innermost




We number the DPs as follows:
  1. H(g(x), y, z) -> H(x, y, z)
  2. H(g(x), y, z) -> F(y, h(x, y, z))
  3. F(g(x), g(y)) -> H(g(y), x, g(y))
  4. F(g(x), a) -> F(x, g(a))
and get the following Size-Change Graph(s):
{1} , {1}
1>1
2=2
3=3
{2} , {2}
2=1
{3} , {3}
1>2
2=1
2=3
{4} , {4}
1>1

which lead(s) to this/these maximal multigraph(s):
{1} , {1}
1>1
2=2
3=3
{3} , {2}
1>1
{2} , {3}
2>2
{4} , {2}
1>1
{3} , {4}
1>1
{2} , {1}
2>2
{1} , {3}
2>2
{1} , {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:
g(x1) -> g(x1)

We obtain no new DP problems.

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