Term Rewriting System R:

   [x, y]
   even(0) -> true
   even(s(0)) -> false
   even(s(s(x))) -> even(x)
   half(0) -> 0
   half(s(s(x))) -> s(half(x))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> if_times(even(s(x)), s(x), y)
   if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y))
   if_times(false, s(x), y) -> plus(y, times(x, y))

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: 


   EVEN(s(s(x))) -> EVEN(x)
   TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)
   TIMES(s(x), y) -> EVEN(s(x))
   IF_TIMES(true, s(x), y) -> PLUS(times(half(s(x)), y), times(half(s(x)), y))
   IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)
   IF_TIMES(true, s(x), y) -> HALF(s(x))
   IF_TIMES(false, s(x), y) -> PLUS(y, times(x, y))
   IF_TIMES(false, s(x), y) -> TIMES(x, y)
   PLUS(s(x), y) -> PLUS(x, y)
   HALF(s(s(x))) -> HALF(x)

Furthermore, R contains four SCCs.


SCC1:

EVEN(s(s(x))) -> EVEN(x)

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(EVEN(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   EVEN(s(s(x))) -> EVEN(x)



 This transformation is resulting in no new subcycles.


SCC2:

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



 This transformation is resulting in no new subcycles.


SCC3:

HALF(s(s(x))) -> HALF(x)

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(HALF(x_1)) = 1 + x_1

The following Dependency Pairs can be deleted:

   HALF(s(s(x))) -> HALF(x)



 This transformation is resulting in no new subcycles.


SCC4:

IF_TIMES(false, s(x), y) -> TIMES(x, y)
IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)
TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   half(0) -> 0
   half(s(s(x))) -> s(half(x))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(TIMES(x_1, x_2)) = x_1
POL(even(x_1)) = 0
POL(true) = 0
POL(IF_TIMES(x_1, x_2, x_3)) = x_2
POL(half(x_1)) = x_1
POL(0) = 0
POL(false) = 0

 resulting in one subcycle.


SCC4.Polo1:

TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)
IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)

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

   TIMES(s(x), y) -> IF_TIMES(even(s(x)), s(x), y)
two new Dependency Pairs
are created:


   TIMES(s(0), y) -> IF_TIMES(false, s(0), y)
   TIMES(s(s(x'')), y) -> IF_TIMES(even(x''), s(s(x'')), y)

The transformation is resulting in one subcycle:


SCC4.Polo1.Nar1:

TIMES(s(s(x'')), y) -> IF_TIMES(even(x''), s(s(x'')), y)
IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)

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

   IF_TIMES(true, s(x), y) -> TIMES(half(s(x)), y)
one new Dependency Pair
is created:


   IF_TIMES(true, s(s(x'')), y) -> TIMES(s(half(x'')), y)

The transformation is resulting in one subcycle:


SCC4.Polo1.Nar1.Nar1:

IF_TIMES(true, s(s(x'')), y) -> TIMES(s(half(x'')), y)
TIMES(s(s(x'')), y) -> IF_TIMES(even(x''), s(s(x'')), y)

By using a polynomial ordering, at least one Dependency Pair of this SCC can be strictly oriented.

Additionally, the following rules can be oriented: 

   half(0) -> 0
   half(s(s(x))) -> s(half(x))


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(TIMES(x_1, x_2)) = x_1
POL(even(x_1)) = 0
POL(true) = 0
POL(IF_TIMES(x_1, x_2, x_3)) = x_2
POL(half(x_1)) = x_1
POL(0) = 0
POL(false) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 11.136 seconds.