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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x), y, y) -> F(x, x, y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Usable Rules (Innermost)


Dependency Pair:

F(g(x), y, y) -> F(x, x, y)


Rule:


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


Strategy:

innermost




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


   R
     ↳DPs
       →DP Problem 1
         ↳UsableRules
           →DP Problem 2
             ↳Size-Change Principle


Dependency Pair:

F(g(x), y, y) -> F(x, x, y)


Rule:

none


Strategy:

innermost




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

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

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.

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