Term Rewriting System R:
[x, y, v, w, z]
sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

SORT(cons(x, y)) -> INSERT(x, sort(y))
SORT(cons(x, y)) -> SORT(y)
INSERT(x, cons(v, w)) -> CHOOSE(x, cons(v, w), x, v)
CHOOSE(x, cons(v, w), 0, s(z)) -> INSERT(x, w)
CHOOSE(x, cons(v, w), s(y), s(z)) -> CHOOSE(x, cons(v, w), y, z)

Furthermore, R contains two SCCs. 


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


Dependency Pairs:

CHOOSE(x, cons(v, w), s(y), s(z)) -> CHOOSE(x, cons(v, w), y, z)
CHOOSE(x, cons(v, w), 0, s(z)) -> INSERT(x, w)
INSERT(x, cons(v, w)) -> CHOOSE(x, cons(v, w), x, v)


Rules:


sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)





The following dependency pair can be strictly oriented:

CHOOSE(x, cons(v, w), 0, s(z)) -> INSERT(x, w)


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(CHOOSE(x1, x2, x3, x4))=  x2  
  POL(0)=  0  
  POL(cons(x1, x2))=  1 + x2  
  POL(INSERT(x1, x2))=  x2  
  POL(s(x1))=  0  

resulting in one new DP problem. 



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


Dependency Pairs:

CHOOSE(x, cons(v, w), s(y), s(z)) -> CHOOSE(x, cons(v, w), y, z)
INSERT(x, cons(v, w)) -> CHOOSE(x, cons(v, w), x, v)


Rules:


sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)





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


   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 3
             ↳DGraph
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Polo


Dependency Pair:

CHOOSE(x, cons(v, w), s(y), s(z)) -> CHOOSE(x, cons(v, w), y, z)


Rules:


sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)





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

CHOOSE(x, cons(v, w), s(y), s(z)) -> CHOOSE(x, cons(v, w), y, z)
one new Dependency Pair is created:

CHOOSE(x', cons(v', w'), s(s(y'')), s(s(z''))) -> CHOOSE(x', cons(v', w'), s(y''), s(z''))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 3
             ↳DGraph
             ...
               →DP Problem 5
                 ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Polo


Dependency Pair:

CHOOSE(x', cons(v', w'), s(s(y'')), s(s(z''))) -> CHOOSE(x', cons(v', w'), s(y''), s(z''))


Rules:


sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)





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

CHOOSE(x', cons(v', w'), s(s(y'')), s(s(z''))) -> CHOOSE(x', cons(v', w'), s(y''), s(z''))
one new Dependency Pair is created:

CHOOSE(x''', cons(v'', w''), s(s(s(y''''))), s(s(s(z'''')))) -> CHOOSE(x''', cons(v'', w''), s(s(y'''')), s(s(z'''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 3
             ↳DGraph
             ...
               →DP Problem 6
                 ↳Polynomial Ordering
       →DP Problem 2
         ↳Polo


Dependency Pair:

CHOOSE(x''', cons(v'', w''), s(s(s(y''''))), s(s(s(z'''')))) -> CHOOSE(x''', cons(v'', w''), s(s(y'''')), s(s(z'''')))


Rules:


sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)





The following dependency pair can be strictly oriented:

CHOOSE(x''', cons(v'', w''), s(s(s(y''''))), s(s(s(z'''')))) -> CHOOSE(x''', cons(v'', w''), s(s(y'''')), s(s(z'''')))


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(CHOOSE(x1, x2, x3, x4))=  1 + x1 + x3  
  POL(cons(x1, x2))=  0  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 3
             ↳DGraph
             ...
               →DP Problem 7
                 ↳Dependency Graph
       →DP Problem 2
         ↳Polo


Dependency Pair:


Rules:


sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

SORT(cons(x, y)) -> SORT(y)


Rules:


sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)





The following dependency pair can be strictly oriented:

SORT(cons(x, y)) -> SORT(y)


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(cons(x1, x2))=  1 + x2  
  POL(SORT(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳Polo
           →DP Problem 8
             ↳Dependency Graph


Dependency Pair:


Rules:


sort(nil) -> nil
sort(cons(x, y)) -> insert(x, sort(y))
insert(x, nil) -> cons(x, nil)
insert(x, cons(v, w)) -> choose(x, cons(v, w), x, v)
choose(x, cons(v, w), y, 0) -> cons(x, cons(v, w))
choose(x, cons(v, w), 0, s(z)) -> cons(v, insert(x, w))
choose(x, cons(v, w), s(y), s(z)) -> choose(x, cons(v, w), y, z)





Using the Dependency Graph resulted in no new DP problems.

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