Term Rewriting System R:

   [z, x, y]
   +(a, b) -> +(b, a)
   +(a, +(b, z)) -> +(b, +(a, z))
   +(+(x, y), z) -> +(x, +(y, z))
   f(a, y) -> a
   f(b, y) -> b
   f(+(x, y), z) -> +(f(x, z), f(y, z))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   F(+(x, y), z) -> +'(f(x, z), f(y, z))
   F(+(x, y), z) -> F(x, z)
   F(+(x, y), z) -> F(y, z)
   +'(a, b) -> +'(b, a)
   +'(a, +(b, z)) -> +'(b, +(a, z))
   +'(a, +(b, z)) -> +'(a, z)
   +'(+(x, y), z) -> +'(x, +(y, z))
   +'(+(x, y), z) -> +'(y, z)

Furthermore, R contains three SCCs.


SCC1:

+'(a, +(b, z)) -> +'(a, 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(b) = 1
POL(a) = 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:

   +'(a, +(b, z)) -> +'(a, z)



 This transformation is resulting in no new subcycles.


SCC2:

+'(+(x, y), z) -> +'(y, z)
+'(+(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(b) = 0
POL(a) = 0
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: 

   f(+(x, y), z) -> +(f(x, z), f(y, z))
   f(b, y) -> b
   f(a, y) -> a


 This transformation is resulting in one new subcycle:


SCC2.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(b) = 0
POL(a) = 0
POL(+'(x_1, x_2)) = 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:

+'(+(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(b) = 0
POL(a) = 0
POL(+(x_1, x_2)) = 1 + x_1
POL(+'(x_1, x_2)) = x_1

 resulting in no subcycles.


SCC3:

F(+(x, y), z) -> F(y, z)
F(+(x, y), z) -> F(x, 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(F(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:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.651 seconds.