Term Rewriting System R:

   [X, Y]
   p(0) -> 0
   p(s(X)) -> X
   leq(0, Y) -> true
   leq(s(X), 0) -> false
   leq(s(X), s(Y)) -> leq(X, Y)
   if(true, X, Y) -> activate(X)
   if(false, X, Y) -> activate(Y)
   diff(X, Y) -> if(leq(X, Y), n__0, n__s(diff(p(X), Y)))
   0 -> n__0
   s(X) -> n__s(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(X)
   activate(X) -> X

Termination of R to be shown.


R contains the following Dependency Pairs: 


   LEQ(s(X), s(Y)) -> LEQ(X, Y)
   ACTIVATE(n__0) -> 0'
   ACTIVATE(n__s(X)) -> S(X)
   IF(false, X, Y) -> ACTIVATE(Y)
   IF(true, X, Y) -> ACTIVATE(X)
   DIFF(X, Y) -> IF(leq(X, Y), n__0, n__s(diff(p(X), Y)))
   DIFF(X, Y) -> LEQ(X, Y)
   DIFF(X, Y) -> DIFF(p(X), Y)
   DIFF(X, Y) -> P(X)

Furthermore, R contains two SCCs.


SCC1:

LEQ(s(X), s(Y)) -> LEQ(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(LEQ(x_1, x_2)) = 1 + x_1 + x_2

The following Dependency Pairs can be deleted:

   LEQ(s(X), s(Y)) -> LEQ(X, Y)



 This transformation is resulting in no new subcycles.


SCC2:

DIFF(X, Y) -> DIFF(p(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(n__0) = 0
POL(0) = 0
POL(p(x_1)) = x_1
POL(DIFF(x_1, x_2)) = 1 + x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   leq(s(X), s(Y)) -> leq(X, Y)
   leq(0, Y) -> true
   leq(s(X), 0) -> false
   s(X) -> n__s(X)
   activate(n__0) -> 0
   activate(n__s(X)) -> s(X)
   activate(X) -> X
   if(false, X, Y) -> activate(Y)
   if(true, X, Y) -> activate(X)
   diff(X, Y) -> if(leq(X, Y), n__0, n__s(diff(p(X), Y)))
   p(s(X)) -> X


 This transformation is resulting in one new subcycle:


SCC2.MRR1:

DIFF(X, Y) -> DIFF(p(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(n__0) = 0
POL(0) = 1
POL(p(x_1)) = x_1
POL(DIFF(x_1, x_2)) = x_1 + x_2

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   0 -> n__0


 This transformation is resulting in one new subcycle:


SCC2.MRR1.MRR1:

DIFF(X, Y) -> DIFF(p(X), Y)

Applying the non-overlappingness check to the DPs.

The transformation is resulting in one subcycle:

SCC2.MRR1.MRR1.NOC1:

DIFF(X, Y) -> DIFF(p(X), Y)

Found an infinite P-chain over R:
P = DIFF(X, Y) -> DIFF(p(X), Y)
R = [p(0) -> 0]
s = DIFF(X, Y) evaluates to t = DIFF(p(X), Y)
Thus, s starts an infinite reduction as s matches t.


Non-Termination of R could be shown.

Duration: 0.710 seconds.