Term Rewriting System R:
[x, y]
f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))

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(f(0, x), 1) -> F(g(f(x, x)), x)
F(f(0, x), 1) -> F(x, x)
F(g(x), y) -> F(x, y)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

F(f(0, x), 1) -> F(x, x)
F(g(x), y) -> F(x, y)
F(f(0, x), 1) -> F(g(f(x, x)), x)


Rules:


f(f(0, x), 1) -> f(g(f(x, x)), x)
f(g(x), y) -> g(f(x, y))


Strategy:

innermost




We number the DPs as follows:
  1. F(f(0, x), 1) -> F(x, x)
  2. F(g(x), y) -> F(x, y)
  3. F(f(0, x), 1) -> F(g(f(x, x)), x)
and get the following Size-Change Graph(s):
{3, 1} , {3, 1}
1>1
1>2
{2} , {2}
1>1
2=2
{3, 1} , {3, 1}
1>2

which lead(s) to this/these maximal multigraph(s):
{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