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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


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


Dependency Pairs:

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


Rules:


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





The following dependency pair can be strictly oriented:

MINUS(h(x)) -> MINUS(x)


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(MINUS(x1))=  x1  
  POL(h(x1))=  1 + x1  
  POL(f(x1, x2))=  x1 + x2  

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


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


Dependency Pairs:

MINUS(f(x, y)) -> MINUS(x)
MINUS(f(x, y)) -> MINUS(y)


Rules:


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





The following dependency pairs can be strictly oriented:

MINUS(f(x, y)) -> MINUS(x)
MINUS(f(x, y)) -> MINUS(y)


There are no usable rules using the Ce-refinement that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(MINUS(x1))=  x1  
  POL(f(x1, x2))=  1 + x1 + x2  

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


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


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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