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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

A -> G(c)
F(g(X), b) -> F(a, X)
F(g(X), b) -> A

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pair:

F(g(X), b) -> F(a, X)


Rules:


a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(g(X), b) -> F(a, X)
one new Dependency Pair is created:

F(g(X), b) -> F(g(c), X)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Argument Filtering and Ordering


Dependency Pair:

F(g(X), b) -> F(g(c), X)


Rules:


a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(g(X), b) -> F(g(c), X)


The following usable rule for innermost w.r.t. to the AFS can be oriented:

g(a) -> b


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(c)=  0  
  POL(g(x1))=  1 + x1  
  POL(b)=  1  
  POL(a)=  0  
  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) -> g(x1)
a -> a


   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳AFS
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


a -> g(c)
g(a) -> b
f(g(X), b) -> f(a, X)


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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