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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a, a) -> F(a, b)
F(a, b) -> F(s(a), c)
F(s(X), c) -> F(X, c)
F(c, c) -> F(a, a)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Polynomial Ordering


Dependency Pairs:

F(c, c) -> F(a, a)
F(s(X), c) -> F(X, c)
F(a, b) -> F(s(a), c)
F(a, a) -> F(a, b)


Rules:


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





The following dependency pair can be strictly oriented:

F(c, c) -> F(a, a)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem. 



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


Dependency Pairs:

F(s(X), c) -> F(X, c)
F(a, b) -> F(s(a), c)
F(a, a) -> F(a, b)


Rules:


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





Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 3
                 ↳Forward Instantiation Transformation


Dependency Pair:

F(s(X), c) -> F(X, c)


Rules:


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





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

F(s(X), c) -> F(X, c)
one new Dependency Pair is created:

F(s(s(X'')), c) -> F(s(X''), c)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation


Dependency Pair:

F(s(s(X'')), c) -> F(s(X''), c)


Rules:


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





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

F(s(s(X'')), c) -> F(s(X''), c)
one new Dependency Pair is created:

F(s(s(s(X''''))), c) -> F(s(s(X'''')), c)

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 5
                 ↳Polynomial Ordering


Dependency Pair:

F(s(s(s(X''''))), c) -> F(s(s(X'''')), c)


Rules:


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





The following dependency pair can be strictly oriented:

F(s(s(s(X''''))), c) -> F(s(s(X'''')), c)


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(c)=  0  
  POL(s(x1))=  1 + x1  
  POL(F(x1, x2))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 2
             ↳DGraph
             ...
               →DP Problem 6
                 ↳Dependency Graph


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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