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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

G(a) -> G(b)
G(a) -> B
B -> F(a, a)
F(a, a) -> G(d)

Furthermore, R contains one SCC. 


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


Dependency Pair:

G(a) -> G(b)


Rules:


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


Strategy:

innermost




The following dependency pair can be strictly oriented:

G(a) -> G(b)


The following usable rules for innermost can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  x1  
  POL(G(x1))=  x1  
  POL(b)=  0  
  POL(d)=  0  
  POL(a)=  1  
  POL(f)=  0  

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


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


Dependency Pair:


Rules:


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


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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