Term Rewriting System R:

   [x, y, z, l]
   0(#) -> #
   +(x, #) -> x
   +(#, x) -> x
   +(0(x), 0(y)) -> 0(+(x, y))
   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))
   +(+(x, y), z) -> +(x, +(y, z))
   *(#, x) -> #
   *(0(x), y) -> 0(*(x, y))
   *(1(x), y) -> +(0(*(x, y)), y)
   *(*(x, y), z) -> *(x, *(y, z))
   sum(nil) -> 0(#)
   sum(cons(x, l)) -> +(x, sum(l))
   prod(nil) -> 1(#)
   prod(cons(x, l)) -> *(x, prod(l))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   *'(1(x), y) -> +'(0(*(x, y)), y)
   *'(1(x), y) -> 0'(*(x, y))
   *'(1(x), y) -> *'(x, y)
   *'(*(x, y), z) -> *'(x, *(y, z))
   *'(*(x, y), z) -> *'(y, z)
   *'(0(x), y) -> 0'(*(x, y))
   *'(0(x), y) -> *'(x, y)
   +'(0(x), 1(y)) -> +'(x, y)
   +'(1(x), 0(y)) -> +'(x, y)
   +'(0(x), 0(y)) -> 0'(+(x, y))
   +'(0(x), 0(y)) -> +'(x, y)
   +'(1(x), 1(y)) -> 0'(+(+(x, y), 1(#)))
   +'(1(x), 1(y)) -> +'(+(x, y), 1(#))
   +'(1(x), 1(y)) -> +'(x, y)
   +'(+(x, y), z) -> +'(x, +(y, z))
   +'(+(x, y), z) -> +'(y, z)
   PROD(cons(x, l)) -> *'(x, prod(l))
   PROD(cons(x, l)) -> PROD(l)
   SUM(nil) -> 0'(#)
   SUM(cons(x, l)) -> +'(x, sum(l))
   SUM(cons(x, l)) -> SUM(l)

Furthermore, R contains four SCCs.


SCC1:

+'(+(x, y), z) -> +'(y, z)
+'(+(x, y), z) -> +'(x, +(y, z))
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(0(x), 0(y)) -> +'(x, y)
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(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(1(x_1)) = x_1
POL(+'(x_1, x_2)) = 1 + x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0(x_1)) = x_1
POL(#) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   *(1(x), y) -> +(0(*(x, y)), y)
   *(*(x, y), z) -> *(x, *(y, z))
   *(0(x), y) -> 0(*(x, y))
   *(#, x) -> #
   prod(cons(x, l)) -> *(x, prod(l))
   prod(nil) -> 1(#)
   sum(nil) -> 0(#)
   sum(cons(x, l)) -> +(x, sum(l))


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

+'(+(x, y), z) -> +'(x, +(y, z))
+'(1(x), 1(y)) -> +'(x, y)
+'(1(x), 1(y)) -> +'(+(x, y), 1(#))
+'(0(x), 0(y)) -> +'(x, y)
+'(1(x), 0(y)) -> +'(x, y)
+'(0(x), 1(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(y, z)

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(1(x_1)) = 1 + x_1
POL(+'(x_1, x_2)) = 1 + x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0(x_1)) = x_1
POL(#) = 0

The following Dependency Pairs can be deleted:

   +'(1(x), 1(y)) -> +'(x, y)
   +'(1(x), 1(y)) -> +'(+(x, y), 1(#))
   +'(1(x), 0(y)) -> +'(x, y)
   +'(0(x), 1(y)) -> +'(x, y)

The following rules of R can be deleted: 

   +(1(x), 1(y)) -> 0(+(+(x, y), 1(#)))


 This transformation is resulting in one new subcycle:


SCC1.MRR1.MRR1:

+'(+(x, y), z) -> +'(y, z)
+'(0(x), 0(y)) -> +'(x, y)
+'(+(x, y), z) -> +'(x, +(y, z))

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(1(x_1)) = x_1
POL(+'(x_1, x_2)) = x_1 + x_2
POL(+(x_1, x_2)) = x_1 + x_2
POL(0(x_1)) = 1 + x_1
POL(#) = 0

The following Dependency Pairs can be deleted:

   +'(0(x), 0(y)) -> +'(x, y)

The following rules of R can be deleted: 

   +(0(x), 1(y)) -> 1(+(x, y))
   +(1(x), 0(y)) -> 1(+(x, y))
   +(0(x), 0(y)) -> 0(+(x, y))
   0(#) -> #
   +(x, #) -> x
   +(#, x) -> x


 This transformation is resulting in one new subcycle:


SCC1.MRR1.MRR1.MRR1:

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

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, y), z) -> +'(y, z)

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC1.MRR1.MRR1.MRR1.MRR1:

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

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

No rules need to be oriented.

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

 resulting in no subcycles.


SCC2:

*'(0(x), y) -> *'(x, y)
*'(*(x, y), z) -> *'(y, z)
*'(*(x, y), z) -> *'(x, *(y, z))
*'(1(x), y) -> *'(x, y)

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(*(x_1, x_2)) = 1 + x_1 + x_2
POL(1(x_1)) = 1 + x_1
POL(*'(x_1, x_2)) = x_1
POL(+(x_1, x_2)) = 0
POL(0(x_1)) = 1 + x_1
POL(#) = 0

 resulting in no 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: 1.210 seconds.