Term Rewriting System R:

   [x, y]
   O(0) -> 0
   +(0, x) -> x
   +(x, 0) -> x
   +(O(x), O(y)) -> O(+(x, y))
   +(O(x), I(y)) -> I(+(x, y))
   +(I(x), O(y)) -> I(+(x, y))
   +(I(x), I(y)) -> O(+(+(x, y), I(0)))
   *(0, x) -> 0
   *(x, 0) -> 0
   *(O(x), y) -> O(*(x, y))
   *(I(x), y) -> +(O(*(x, y)), y)

Termination of R to be shown.


R contains the following Dependency Pairs: 


   *'(O(x), y) -> O'(*(x, y))
   *'(O(x), y) -> *'(x, y)
   *'(I(x), y) -> +'(O(*(x, y)), y)
   *'(I(x), y) -> O'(*(x, y))
   *'(I(x), y) -> *'(x, y)
   +'(I(x), O(y)) -> +'(x, y)
   +'(I(x), I(y)) -> O'(+(+(x, y), I(0)))
   +'(I(x), I(y)) -> +'(+(x, y), I(0))
   +'(I(x), I(y)) -> +'(x, y)
   +'(O(x), O(y)) -> O'(+(x, y))
   +'(O(x), O(y)) -> +'(x, y)
   +'(O(x), I(y)) -> +'(x, y)

Furthermore, R contains two SCCs.


SCC1:

+'(O(x), I(y)) -> +'(x, y)
+'(O(x), O(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(I(x), O(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(I(x_1)) = x_1
POL(O(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) = 0

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   *(O(x), y) -> O(*(x, y))
   *(0, x) -> 0
   *(x, 0) -> 0
   *(I(x), y) -> +(O(*(x, y)), y)


 This transformation is resulting in one new subcycle:


SCC1.MRR1:

+'(O(x), O(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(x, y)
+'(I(x), I(y)) -> +'(+(x, y), I(0))
+'(I(x), O(y)) -> +'(x, y)
+'(O(x), I(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(I(x_1)) = 1 + x_1
POL(O(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) = 0

The following Dependency Pairs can be deleted:

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

The following rules of R can be deleted: 

   +(I(x), I(y)) -> O(+(+(x, y), I(0)))


 This transformation is resulting in one new subcycle:


SCC1.MRR1.MRR1:

+'(O(x), O(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(O(x_1)) = 1 + x_1
POL(+'(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

*'(I(x), y) -> *'(x, y)
*'(O(x), 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(I(x_1)) = 1 + x_1
POL(O(x_1)) = 1 + x_1
POL(*'(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   *'(I(x), y) -> *'(x, y)
   *'(O(x), y) -> *'(x, y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.910 seconds.