Term Rewriting System R:

   [x, y, l]
   +(x, 0) -> x
   +(0, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   *(x, 0) -> 0
   *(0, x) -> 0
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
   sum(nil) -> 0
   sum(cons(x, l)) -> +(x, sum(l))
   prod(nil) -> s(0)
   prod(cons(x, l)) -> *(x, prod(l))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   +'(s(x), s(y)) -> +'(x, y)
   *'(s(x), s(y)) -> +'(*(x, y), +(x, y))
   *'(s(x), s(y)) -> *'(x, y)
   *'(s(x), s(y)) -> +'(x, y)
   PROD(cons(x, l)) -> *'(x, prod(l))
   PROD(cons(x, l)) -> PROD(l)
   SUM(cons(x, l)) -> +'(x, sum(l))
   SUM(cons(x, l)) -> SUM(l)

Furthermore, R contains four SCCs.


SCC1:

+'(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.


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:

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



 This transformation is resulting in no new subcycles.


SCC4:

PROD(cons(x, l)) -> PROD(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(PROD(x_1)) = 1 + x_1
POL(cons(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   PROD(cons(x, l)) -> PROD(l)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.983 seconds.