Term Rewriting System R:

   [y, x, ys, xs]
   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   ++(nil, ys) -> ys
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   sum(:(x, nil)) -> :(x, nil)
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   -(x, 0) -> x
   -(0, s(y)) -> 0
   -(s(x), s(y)) -> -(x, y)
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   hd(:(x, xs)) -> x
   avg(xs) -> quot(hd(sum(xs)), length(xs))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   +'(s(x), y) -> +'(x, y)
   QUOT(s(x), s(y)) -> QUOT(-(x, y), s(y))
   QUOT(s(x), s(y)) -> -'(x, y)
   -'(s(x), s(y)) -> -'(x, y)
   AVG(xs) -> QUOT(hd(sum(xs)), length(xs))
   AVG(xs) -> HD(sum(xs))
   AVG(xs) -> SUM(xs)
   AVG(xs) -> LENGTH(xs)
   SUM(:(x, :(y, xs))) -> SUM(:(+(x, y), xs))
   SUM(:(x, :(y, xs))) -> +'(x, y)
   SUM(++(xs, :(x, :(y, ys)))) -> SUM(++(xs, sum(:(x, :(y, ys)))))
   SUM(++(xs, :(x, :(y, ys)))) -> ++'(xs, sum(:(x, :(y, ys))))
   SUM(++(xs, :(x, :(y, ys)))) -> SUM(:(x, :(y, ys)))
   LENGTH(:(x, xs)) -> LENGTH(xs)
   ++'(:(x, xs), ys) -> ++'(xs, ys)

Furthermore, R contains seven SCCs.


SCC1:

+'(s(x), y) -> +'(x, y)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(+'(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   +'(s(x), y) -> +'(x, y)



 This transformation is resulting in no new subcycles.


SCC2:

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

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(-'(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC3:

++'(:(x, xs), ys) -> ++'(xs, ys)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(++'(x_1, x_2)) = 1 + x_1 + x_2
POL(:(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   ++'(:(x, xs), ys) -> ++'(xs, ys)



 This transformation is resulting in no new subcycles.


SCC4:

LENGTH(:(x, xs)) -> LENGTH(xs)

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(LENGTH(x_1)) = 1 + x_1
POL(:(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   LENGTH(:(x, xs)) -> LENGTH(xs)



 This transformation is resulting in no new subcycles.


SCC5:

SUM(:(x, :(y, xs))) -> SUM(:(+(x, y), xs))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = x_1
POL(SUM(x_1)) = 1 + x_1
POL(:(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0) = 1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   hd(:(x, xs)) -> x
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   quot(0, s(y)) -> 0
   -(0, s(y)) -> 0
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   avg(xs) -> quot(hd(sum(xs)), length(xs))
   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))
   sum(++(xs, :(x, :(y, ys)))) -> sum(++(xs, sum(:(x, :(y, ys)))))
   sum(:(x, nil)) -> :(x, nil)
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   ++(:(x, xs), ys) -> :(x, ++(xs, ys))
   ++(nil, ys) -> ys
   +(0, y) -> y


 This transformation is resulting in one new subcycle:


SCC5.MRR1:

SUM(:(x, :(y, xs))) -> SUM(:(+(x, y), xs))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(s(x_1)) = x_1
POL(SUM(x_1)) = 1 + x_1
POL(:(x_1, x_2)) = 1 + x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2

The following Dependency Pairs can be deleted:

   SUM(:(x, :(y, xs))) -> SUM(:(+(x, y), xs))



 This transformation is resulting in no new subcycles.


SCC6:

SUM(++(xs, :(x, :(y, ys)))) -> SUM(++(xs, sum(:(x, :(y, ys)))))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(SUM(x_1)) = 1 + x_1
POL(sum(x_1)) = x_1
POL(:(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0) = 1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   hd(:(x, xs)) -> x
   quot(s(x), s(y)) -> s(quot(-(x, y), s(y)))
   quot(0, s(y)) -> 0
   -(0, s(y)) -> 0
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   avg(xs) -> quot(hd(sum(xs)), length(xs))
   length(nil) -> 0
   length(:(x, xs)) -> s(length(xs))
   +(0, y) -> y


 This transformation is resulting in one new subcycle:


SCC6.MRR1:

SUM(++(xs, :(x, :(y, ys)))) -> SUM(++(xs, sum(:(x, :(y, ys)))))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = x_1
POL(SUM(x_1)) = 1 + x_1
POL(sum(x_1)) = x_1
POL(:(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   ++(nil, ys) -> ys


 This transformation is resulting in one new subcycle:


SCC6.MRR1.MRR1:

SUM(++(xs, :(x, :(y, ys)))) -> SUM(++(xs, sum(:(x, :(y, ys)))))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(SUM(x_1)) = 1 + x_1
POL(sum(x_1)) = x_1
POL(:(x_1, x_2)) = 1 + x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   sum(:(x, :(y, xs))) -> sum(:(+(x, y), xs))


 This transformation is resulting in one new subcycle:


SCC6.MRR1.MRR1.MRR1:

SUM(++(xs, :(x, :(y, ys)))) -> SUM(++(xs, sum(:(x, :(y, ys)))))

Removing rules from R by ordering and analyzing Dependency Pairs, Usable Rules, and Usable Equations.

This is possible by using the following (C_E-compatible) Polynomial ordering.

Polynomial interpretation:
POL(nil) = 0
POL(++(x_1, x_2)) = x_1 + x_2
POL(SUM(x_1)) = 1 + x_1
POL(sum(x_1)) = x_1
POL(:(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   +(s(x), y) -> s(+(x, y))


 This transformation is resulting in one new subcycle:


SCC6.MRR1.MRR1.MRR1.MRR1:

SUM(++(xs, :(x, :(y, ys)))) -> SUM(++(xs, sum(:(x, :(y, ys)))))

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

   SUM(++(xs, :(x, :(y, ys)))) -> SUM(++(xs, sum(:(x, :(y, ys)))))
one new Dependency Pair
is created:


   SUM(++(:(x'', xs''), :(x0, :(y', ys'')))) -> SUM(:(x'', ++(xs'', sum(:(x0, :(y', ys''))))))

The transformation is resulting in no subcycles.


SCC7:

QUOT(s(x), s(y)) -> QUOT(-(x, y), s(y))

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   -(0, s(y)) -> 0
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(-(x_1, x_2)) = x_1
POL(QUOT(x_1, x_2)) = 1 + x_1
POL(0) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 1.152 seconds.