Term Rewriting System R:

   [y, x]
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   pred(s(x)) -> x
   minus(x, 0) -> x
   minus(x, s(y)) -> pred(minus(x, y))
   mod(0, y) -> 0
   mod(s(x), 0) -> 0
   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
   if_mod(false, s(x), s(y)) -> s(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: 


   LE(s(x), s(y)) -> LE(x, y)
   MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(y))
   MOD(s(x), s(y)) -> LE(y, x)
   IF_MOD(true, s(x), s(y)) -> MOD(minus(x, y), s(y))
   IF_MOD(true, s(x), s(y)) -> MINUS(x, y)
   MINUS(x, s(y)) -> PRED(minus(x, y))
   MINUS(x, s(y)) -> MINUS(x, y)

Furthermore, R contains three 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, 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.


SCC3:

IF_MOD(true, s(x), s(y)) -> MOD(minus(x, y), s(y))
MOD(s(x), s(y)) -> IF_MOD(le(y, x), s(x), s(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: 

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


Used ordering: Polynomial ordering with Polynomial interpretation:
POL(IF_MOD(x_1, x_2, x_3)) = x_2
POL(s(x_1)) = 1 + x_1
POL(le(x_1, x_2)) = 0
POL(MOD(x_1, x_2)) = x_1
POL(minus(x_1, x_2)) = x_1
POL(pred(x_1)) = x_1
POL(true) = 0
POL(0) = 0
POL(false) = 0

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.852 seconds.