Term Rewriting System R:

   [x, y]
   times(x, plus(y, 1)) -> plus(times(x, plus(y, times(1, 0))), x)
   times(x, 1) -> x
   times(x, 0) -> 0
   plus(x, 0) -> x

Termination of R to be shown.


This program has no overlaps, so it is sufficient to show innermost termination. 


R contains the following Dependency Pairs: 


   TIMES(x, plus(y, 1)) -> PLUS(times(x, plus(y, times(1, 0))), x)
   TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, times(1, 0)))
   TIMES(x, plus(y, 1)) -> PLUS(y, times(1, 0))
   TIMES(x, plus(y, 1)) -> TIMES(1, 0)

Furthermore, R contains one SCC.


SCC1:

TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, times(1, 0)))

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

   TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, times(1, 0)))
one new Dependency Pair
is created:


   TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, 0))

The transformation is resulting in one subcycle:


SCC1.Nar1:

TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, 0))

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

   TIMES(x, plus(y, 1)) -> TIMES(x, plus(y, 0))
one new Dependency Pair
is created:


   TIMES(x, plus(y, 1)) -> TIMES(x, y)

The transformation is resulting in one subcycle:


SCC1.Nar1.Rewr1:

TIMES(x, plus(y, 1)) -> 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)) = 1 + x_1 + x_2
POL(1) = 1
POL(TIMES(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   TIMES(x, plus(y, 1)) -> TIMES(x, y)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 10.617 seconds.