Term Rewriting System R:

   [x, y, u, z]
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   <=(0, y) -> true
   <=(s(x), 0) -> false
   <=(s(x), s(y)) -> <=(x, y)
   if(true, x, y) -> x
   if(false, x, y) -> y
   perfectp(0) -> false
   perfectp(s(x)) -> f(x, s(0), s(x), s(x))
   f(0, y, 0, u) -> true
   f(0, y, s(z), u) -> false
   f(s(x), 0, z, u) -> f(x, u, -(z, s(x)), u)
   f(s(x), s(y), z, u) -> if(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))

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: 


   F(s(x), 0, z, u) -> F(x, u, -(z, s(x)), u)
   F(s(x), 0, z, u) -> -'(z, s(x))
   F(s(x), s(y), z, u) -> IF(<=(x, y), f(s(x), -(y, x), z, u), f(x, u, z, u))
   F(s(x), s(y), z, u) -> <='(x, y)
   F(s(x), s(y), z, u) -> F(s(x), -(y, x), z, u)
   F(s(x), s(y), z, u) -> -'(y, x)
   F(s(x), s(y), z, u) -> F(x, u, z, u)
   -'(s(x), s(y)) -> -'(x, y)
   <='(s(x), s(y)) -> <='(x, y)
   PERFECTP(s(x)) -> F(x, s(0), s(x), s(x))

Furthermore, R contains three SCCs.


SCC1:

-'(s(x), s(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(s(x_1)) = 1 + x_1
POL(-'(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

<='(s(x), s(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(s(x_1)) = 1 + x_1
POL(<='(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   <='(s(x), s(y)) -> <='(x, y)



 This transformation is resulting in no new subcycles.


SCC3:

F(s(x), s(y), z, u) -> F(x, u, z, u)
F(s(x), s(y), z, u) -> F(s(x), -(y, x), z, u)
F(s(x), 0, z, u) -> F(x, u, -(z, s(x)), u)

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

No rules need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(-(x_1, x_2)) = 0
POL(F(x_1, x_2, x_3, x_4)) = x_1
POL(0) = 0

 resulting in one subcycle.


SCC3.Polo1:

F(s(x), s(y), z, u) -> F(s(x), -(y, x), z, u)

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

Additionally, the following rules can be oriented: 

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(s(x_1)) = 1 + x_1
POL(-(x_1, x_2)) = x_1
POL(F(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
POL(0) = 1

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.994 seconds.