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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a, y) -> F(y, g(y))
F(a, y) -> G(y)

Furthermore, R contains one SCC. 


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


Dependency Pair:

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


Rules:


f(a, y) -> f(y, g(y))
g(a) -> b
g(b) -> b


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


The following usable rules for innermost can be oriented:

g(a) -> b
g(b) -> b


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g)=  0  
  POL(b)=  0  
  POL(a)=  1  
  POL(F(x1, x2))=  x1 + x2  

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


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


Dependency Pair:


Rules:


f(a, y) -> f(y, g(y))
g(a) -> b
g(b) -> b


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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