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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

H(f(x), y) -> G(x, y)
G(x, y) -> H(x, y)

Furthermore, R contains one SCC. 


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


Dependency Pairs:

G(x, y) -> H(x, y)
H(f(x), y) -> G(x, y)


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(x, y) -> H(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(H(x1, x2))=  x1 + x2  
  POL(f(x1))=  1 + x1  

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


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


Dependency Pair:

H(f(x), y) -> G(x, y)


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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