Term Rewriting System R:

   [x, y, z, k, l]
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(0, s(y)) -> 0
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   app(nil, k) -> k
   app(l, nil) -> l
   app(cons(x, l), k) -> cons(x, app(l, k))
   sum(cons(x, nil)) -> cons(x, nil)
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))
   SUM(app(l, cons(x, cons(y, k)))) -> APP(l, sum(cons(x, cons(y, k))))
   SUM(app(l, cons(x, cons(y, k)))) -> SUM(cons(x, cons(y, k)))
   SUM(cons(x, cons(y, l))) -> SUM(cons(plus(x, y), l))
   SUM(cons(x, cons(y, l))) -> PLUS(x, y)
   PLUS(s(x), y) -> PLUS(x, y)
   MINUS(s(x), s(y)) -> MINUS(x, y)
   MINUS(minus(x, y), z) -> MINUS(x, plus(y, z))
   MINUS(minus(x, y), z) -> PLUS(y, z)
   APP(cons(x, l), k) -> APP(l, k)
   QUOT(s(x), s(y)) -> QUOT(minus(x, y), s(y))
   QUOT(s(x), s(y)) -> MINUS(x, y)

Furthermore, R contains six SCCs.


SCC1:

APP(cons(x, l), k) -> APP(l, k)

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(APP(x_1, x_2)) = 1 + x_1 + x_2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   APP(cons(x, l), k) -> APP(l, k)



 This transformation is resulting in no new subcycles.


SCC2:

PLUS(s(x), y) -> PLUS(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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC3:

SUM(cons(x, cons(y, l))) -> SUM(cons(plus(x, y), l))

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(plus(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(SUM(x_1)) = 1 + x_1
POL(0) = 1
POL(cons(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   sum(cons(x, nil)) -> cons(x, nil)
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   app(cons(x, l), k) -> cons(x, app(l, k))
   app(l, nil) -> l
   app(nil, k) -> k
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   quot(0, s(y)) -> 0
   plus(0, y) -> y


 This transformation is resulting in one new subcycle:


SCC3.MRR1:

SUM(cons(x, cons(y, l))) -> SUM(cons(plus(x, y), l))

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(plus(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(SUM(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   SUM(cons(x, cons(y, l))) -> SUM(cons(plus(x, y), l))



 This transformation is resulting in no new subcycles.


SCC4:

SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))

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(plus(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(app(x_1, x_2)) = x_1 + x_2
POL(0) = 1
POL(cons(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   quot(0, s(y)) -> 0
   plus(0, y) -> y


 This transformation is resulting in one new subcycle:


SCC4.MRR1:

SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))

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(plus(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(app(x_1, x_2)) = 1 + x_1 + x_2
POL(cons(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   app(l, nil) -> l
   app(nil, k) -> k


 This transformation is resulting in one new subcycle:


SCC4.MRR1.MRR1:

SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))

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(plus(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(app(x_1, x_2)) = x_1 + x_2
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))


 This transformation is resulting in one new subcycle:


SCC4.MRR1.MRR1.MRR1:

SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))

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(SUM(x_1)) = 1 + x_1
POL(sum(x_1)) = x_1
POL(app(x_1, x_2)) = x_1 + x_2
POL(cons(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

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


 This transformation is resulting in one new subcycle:


SCC4.MRR1.MRR1.MRR1.MRR1:

SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))

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

   SUM(app(l, cons(x, cons(y, k)))) -> SUM(app(l, sum(cons(x, cons(y, k)))))
one new Dependency Pair
is created:


   SUM(app(cons(x'', l''), cons(x0, cons(y', k'')))) -> SUM(cons(x'', app(l'', sum(cons(x0, cons(y', k''))))))

The transformation is resulting in no subcycles.


SCC5:

MINUS(minus(x, y), z) -> MINUS(x, plus(y, z))
MINUS(s(x), s(y)) -> MINUS(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(plus(x_1, x_2)) = x_1 + x_2
POL(s(x_1)) = x_1
POL(minus(x_1, x_2)) = x_1 + x_2
POL(MINUS(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 1

The following Dependency Pairs can be deleted:

   MINUS(minus(x, y), z) -> MINUS(x, plus(y, z))

The following rules of R can be deleted: 

   sum(cons(x, nil)) -> cons(x, nil)
   sum(app(l, cons(x, cons(y, k)))) -> sum(app(l, sum(cons(x, cons(y, k)))))
   sum(cons(x, cons(y, l))) -> sum(cons(plus(x, y), l))
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))
   app(cons(x, l), k) -> cons(x, app(l, k))
   app(l, nil) -> l
   app(nil, k) -> k
   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
   quot(0, s(y)) -> 0
   plus(0, y) -> y


 This transformation is resulting in one new subcycle:


SCC5.MRR1:

MINUS(s(x), s(y)) -> MINUS(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(MINUS(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC6:

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

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   minus(minus(x, y), z) -> minus(x, plus(y, z))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(plus(x_1, x_2)) = 0
POL(minus(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: 1.65 seconds.