Term Rewriting System R:

   [x, y, z, l, l1, l2]
   +(x, 0) -> x
   +(0, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   +(+(x, y), z) -> +(x, +(y, z))
   *(x, 0) -> 0
   *(0, x) -> 0
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
   *(*(x, y), z) -> *(x, *(y, z))
   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   sum(nil) -> 0
   sum(cons(x, l)) -> +(x, sum(l))
   sum(app(l1, l2)) -> +(sum(l1), sum(l2))
   prod(nil) -> s(0)
   prod(cons(x, l)) -> *(x, prod(l))
   prod(app(l1, l2)) -> *(prod(l1), prod(l2))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   APP(cons(x, l1), l2) -> APP(l1, l2)
   +'(s(x), s(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)
   PROD(app(l1, l2)) -> *'(prod(l1), prod(l2))
   PROD(app(l1, l2)) -> PROD(l1)
   PROD(app(l1, l2)) -> PROD(l2)
   *'(s(x), s(y)) -> +'(*(x, y), +(x, y))
   *'(s(x), s(y)) -> *'(x, y)
   *'(s(x), s(y)) -> +'(x, y)
   *'(*(x, y), z) -> *'(x, *(y, z))
   *'(*(x, y), z) -> *'(y, z)
   SUM(cons(x, l)) -> +'(x, sum(l))
   SUM(cons(x, l)) -> SUM(l)
   SUM(app(l1, l2)) -> +'(sum(l1), sum(l2))
   SUM(app(l1, l2)) -> SUM(l1)
   SUM(app(l1, l2)) -> SUM(l2)

Furthermore, R contains five SCCs.


SCC1:

APP(cons(x, l1), l2) -> APP(l1, l2)

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, l1), l2) -> APP(l1, l2)



 This transformation is resulting in no new subcycles.


SCC2:

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

   app(nil, l) -> l
   app(cons(x, l1), l2) -> cons(x, app(l1, l2))
   prod(cons(x, l)) -> *(x, prod(l))
   prod(app(l1, l2)) -> *(prod(l1), prod(l2))
   prod(nil) -> s(0)
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
   *(*(x, y), z) -> *(x, *(y, z))
   *(0, x) -> 0
   *(x, 0) -> 0
   sum(cons(x, l)) -> +(x, sum(l))
   sum(nil) -> 0
   sum(app(l1, l2)) -> +(sum(l1), sum(l2))
   +(0, x) -> x
   +(x, 0) -> x


 This transformation is resulting in one new subcycle:


SCC2.MRR1:

+'(+(x, y), z) -> +'(x, +(y, z))
+'(s(x), s(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(s(x_1)) = x_1
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:


SCC2.MRR1.MRR1:

+'(s(x), s(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(s(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:

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

No rules of R can be deleted.


 This transformation is resulting in one new subcycle:


SCC2.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(s(x_1)) = 0
POL(+(x_1, x_2)) = 1 + x_1
POL(+'(x_1, x_2)) = x_1

 resulting in no subcycles.


SCC3:

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

 resulting in no subcycles.


SCC4:

SUM(app(l1, l2)) -> SUM(l2)
SUM(app(l1, l2)) -> SUM(l1)
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(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:

   SUM(app(l1, l2)) -> SUM(l2)
   SUM(app(l1, l2)) -> SUM(l1)
   SUM(cons(x, l)) -> SUM(l)



 This transformation is resulting in no new subcycles.


SCC5:

PROD(app(l1, l2)) -> PROD(l2)
PROD(app(l1, l2)) -> PROD(l1)
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(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:

   PROD(app(l1, l2)) -> PROD(l2)
   PROD(app(l1, l2)) -> PROD(l1)
   PROD(cons(x, l)) -> PROD(l)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 1.29 seconds.