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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pair:

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


Rule:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules for innermost that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1, x2))=  1 + x1 + x2  
  POL(F(x1, x2))=  1 + x1 + x2  
  POL(f(x1, x2))=  1 + x1 + x2  

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


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


Dependency Pair:


Rule:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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