Term Rewriting System R:

   [X, Y]
   minus(X, s(Y)) -> pred(minus(X, Y))
   minus(X, 0) -> X
   pred(s(X)) -> X
   le(s(X), s(Y)) -> le(X, Y)
   le(s(X), 0) -> false
   le(0, Y) -> true
   gcd(0, Y) -> 0
   gcd(s(X), 0) -> s(X)
   gcd(s(X), s(Y)) -> if(le(Y, X), s(X), s(Y))
   if(true, s(X), s(Y)) -> gcd(minus(X, Y), s(Y))
   if(false, s(X), s(Y)) -> gcd(minus(Y, X), 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: 


   GCD(s(X), s(Y)) -> IF(le(Y, X), s(X), s(Y))
   GCD(s(X), s(Y)) -> LE(Y, X)
   IF(false, s(X), s(Y)) -> GCD(minus(Y, X), s(X))
   IF(false, s(X), s(Y)) -> MINUS(Y, X)
   IF(true, s(X), s(Y)) -> GCD(minus(X, Y), s(Y))
   IF(true, s(X), s(Y)) -> MINUS(X, Y)
   LE(s(X), s(Y)) -> LE(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(true, s(X), s(Y)) -> GCD(minus(X, Y), s(Y))
IF(false, s(X), s(Y)) -> GCD(minus(Y, X), s(X))
GCD(s(X), s(Y)) -> IF(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(s(x_1)) = 1 + x_1
POL(le(x_1, x_2)) = 0
POL(minus(x_1, x_2)) = x_1
POL(pred(x_1)) = x_1
POL(true) = 0
POL(GCD(x_1, x_2)) = x_1 + x_2
POL(0) = 0
POL(false) = 0
POL(IF(x_1, x_2, x_3)) = x_2 + x_3

 resulting in no subcycles.




Termination of R successfully shown.


Duration: 0.949 seconds.