Term Rewriting System R:
[x, y]
-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(-(x, y), -(x, y))
-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(x, y)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Polynomial Ordering


Dependency Pairs:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(x, y)
-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(-(x, y), -(x, y))


Rule:


-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))





The following dependency pair can be strictly oriented:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(x, y)


Additionally, the following usable rule using the Ce-refinement can be oriented:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(-'(x1, x2))=  1 + x1  
  POL(neg(x1))=  x1  
  POL(-(x1, x2))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 2
Polynomial Ordering


Dependency Pair:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(-(x, y), -(x, y))


Rule:


-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))





The following dependency pair can be strictly oriented:

-'(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -'(-(x, y), -(x, y))


Additionally, the following usable rule using the Ce-refinement can be oriented:

-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(-'(x1, x2))=  x1  
  POL(neg(x1))=  1 + x1  
  POL(-(x1, x2))=  1 + x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 2
Polo
             ...
               →DP Problem 3
Dependency Graph


Dependency Pair:


Rule:


-(-(neg(x), neg(x)), -(neg(y), neg(y))) -> -(-(x, y), -(x, y))





Using the Dependency Graph resulted in no new DP problems.

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