Term Rewriting System R:

   [x, y, z]
   times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
   times(x, 0) -> 0
   times(x, s(y)) -> plus(times(x, y), x)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))

Termination of R to be shown.


R contains the following Dependency Pairs: 


   TIMES(x, s(y)) -> PLUS(times(x, y), x)
   TIMES(x, s(y)) -> TIMES(x, y)
   TIMES(x, plus(y, s(z))) -> PLUS(times(x, plus(y, times(s(z), 0))), times(x, s(z)))
   TIMES(x, plus(y, s(z))) -> TIMES(x, plus(y, times(s(z), 0)))
   TIMES(x, plus(y, s(z))) -> PLUS(y, times(s(z), 0))
   TIMES(x, plus(y, s(z))) -> TIMES(s(z), 0)
   TIMES(x, plus(y, s(z))) -> TIMES(x, s(z))
   PLUS(x, s(y)) -> PLUS(x, y)

Furthermore, R contains two SCCs.


SCC1:

PLUS(x, s(y)) -> PLUS(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(PLUS(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   PLUS(x, s(y)) -> PLUS(x, y)



 This transformation is resulting in no new subcycles.


SCC2:

TIMES(x, plus(y, s(z))) -> TIMES(x, s(z))
TIMES(x, plus(y, s(z))) -> TIMES(x, plus(y, times(s(z), 0)))
TIMES(x, s(y)) -> TIMES(x, y)

On this Scc, a Narrowing SCC transformation can be performed.
 
As a result of transforming the rule 

   TIMES(x, plus(y, s(z))) -> TIMES(x, plus(y, times(s(z), 0)))
one new Dependency Pair
is created:


   TIMES(x, plus(y, s(z'))) -> TIMES(x, plus(y, 0))

The transformation is resulting in one subcycle:


SCC2.Nar1:

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

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   times(x, 0) -> 0
   times(x, s(y)) -> plus(times(x, y), x)
   times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z)))


 This transformation is resulting in one new subcycle:


SCC2.Nar1.MRR1:

TIMES(x, plus(y, s(z))) -> TIMES(x, s(z))
TIMES(x, s(y)) -> TIMES(x, y)
TIMES(x, plus(y, s(z'))) -> TIMES(x, plus(y, 0))

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(plus(x_1, x_2)) = 1 + x_1 + x_2
POL(s(x_1)) = x_1
POL(TIMES(x_1, x_2)) = 1 + x_1 + x_2
POL(0) = 0

The following Dependency Pairs can be deleted:

   TIMES(x, plus(y, s(z))) -> TIMES(x, s(z))

The following rules of R can be deleted: 

   plus(x, 0) -> x


 This transformation is resulting in one new subcycle:


SCC2.Nar1.MRR1.MRR1:

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

The following Dependency Pairs can be deleted:

   TIMES(x, plus(y, s(z'))) -> TIMES(x, plus(y, 0))
   TIMES(x, s(y)) -> TIMES(x, y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 10.799 seconds.