Term Rewriting System R:

   [X, Y]
   fact(X) -> if(zero(X), s(0), prod(X, fact(p(X))))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   prod(0, X) -> 0
   prod(s(X), Y) -> add(Y, prod(X, Y))
   if(true, X, Y) -> X
   if(false, X, Y) -> Y
   zero(0) -> true
   zero(s(X)) -> false
   p(s(X)) -> 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: 


   ADD(s(X), Y) -> ADD(X, Y)
   FACT(X) -> IF(zero(X), s(0), prod(X, fact(p(X))))
   FACT(X) -> ZERO(X)
   FACT(X) -> PROD(X, fact(p(X)))
   FACT(X) -> FACT(p(X))
   FACT(X) -> P(X)
   PROD(s(X), Y) -> ADD(Y, prod(X, Y))
   PROD(s(X), Y) -> PROD(X, Y)

Furthermore, R contains three SCCs.


SCC1:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC2:

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

The following Dependency Pairs can be deleted:

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



 This transformation is resulting in no new subcycles.


SCC3:

FACT(X) -> FACT(p(X))

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(FACT(x_1)) = 1 + x_1
POL(p(x_1)) = x_1

No Dependency Pairs can be deleted.

The following rules of R can be deleted: 

   p(s(X)) -> X


 This transformation is resulting in one new subcycle:


SCC3.MRR1:

FACT(X) -> FACT(p(X))

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


Non-Termination of R could be shown.

Duration: 0.659 seconds.