Term Rewriting System R:

   [x, y]
   p(s(x)) -> x
   fact(0) -> s(0)
   fact(s(x)) -> *(s(x), fact(p(s(x))))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

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: 


   +'(x, s(y)) -> +'(x, y)
   *'(s(x), y) -> +'(*(x, y), y)
   *'(s(x), y) -> *'(x, y)
   FACT(s(x)) -> *'(s(x), fact(p(s(x))))
   FACT(s(x)) -> FACT(p(s(x)))
   FACT(s(x)) -> P(s(x))

Furthermore, R contains three SCCs.


SCC1:

+'(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:

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



 This transformation is resulting in no new subcycles.


SCC2:

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



 This transformation is resulting in no new subcycles.


SCC3:

FACT(s(x)) -> FACT(p(s(x)))

On this Scc, a Rewriting SCC transformation can be performed.
 
As a result of transforming the rule 

   FACT(s(x)) -> FACT(p(s(x)))
one new Dependency Pair
is created:


   FACT(s(x)) -> FACT(x)

The transformation is resulting in one subcycle:


SCC3.Rewr1:

FACT(s(x)) -> FACT(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

The following Dependency Pairs can be deleted:

   FACT(s(x)) -> FACT(x)



 This transformation is resulting in no new subcycles.




Termination of R successfully shown.


Duration: 0.571 seconds.