Term Rewriting System R:

   [x, y]
   p(0) -> 0
   p(s(x)) -> x
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y)))))
   if(true, x, y) -> x
   if(false, x, y) -> 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: 


   MINUS(x, s(y)) -> IF(le(x, s(y)), 0, p(minus(x, p(s(y)))))
   MINUS(x, s(y)) -> LE(x, s(y))
   MINUS(x, s(y)) -> P(minus(x, p(s(y))))
   MINUS(x, s(y)) -> MINUS(x, p(s(y)))
   MINUS(x, s(y)) -> P(s(y))
   LE(s(x), s(y)) -> LE(x, y)

Furthermore, R contains two SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

MINUS(x, s(y)) -> MINUS(x, p(s(y)))

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

   MINUS(x, s(y)) -> MINUS(x, p(s(y)))
one new Dependency Pair
is created:


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

The transformation is resulting in one subcycle:


SCC2.Rewr1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.625 seconds.