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
         ↳Argument Filtering and Ordering


Dependency Pair:

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


Rule:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  1 + x1  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2, x3) -> x1
g(x1) -> g(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rule:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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