NO

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S Y:S)
(RULES
activate(n__prod(X1:S,X2:S)) -> prod(X1:S,X2:S)
activate(n__s(X:S)) -> s(X:S)
activate(X:S) -> X:S
add(s(X:S),Y:S) -> s(add(X:S,Y:S))
add(0,X:S) -> X:S
fact(X:S) -> if(zero(X:S),n__s(0),n__prod(X:S,fact(p(X:S))))
if(ffalse,X:S,Y:S) -> activate(Y:S)
if(ttrue,X:S,Y:S) -> activate(X:S)
p(s(X:S)) -> X:S
prod(s(X:S),Y:S) -> add(Y:S,prod(X:S,Y:S))
prod(0,X:S) -> 0
prod(X1:S,X2:S) -> n__prod(X1:S,X2:S)
s(X:S) -> n__s(X:S)
zero(s(X:S)) -> ffalse
zero(0) -> ttrue
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__prod(X1:S,X2:S)) -> PROD(X1:S,X2:S)
 ACTIVATE(n__s(X:S)) -> S(X:S)
 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S)
 ADD(s(X:S),Y:S) -> S(add(X:S,Y:S))
 FACT(X:S) -> FACT(p(X:S))
 FACT(X:S) -> IF(zero(X:S),n__s(0),n__prod(X:S,fact(p(X:S))))
 FACT(X:S) -> P(X:S)
 FACT(X:S) -> ZERO(X:S)
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
 IF(ttrue,X:S,Y:S) -> ACTIVATE(X:S)
 PROD(s(X:S),Y:S) -> ADD(Y:S,prod(X:S,Y:S))
 PROD(s(X:S),Y:S) -> PROD(X:S,Y:S)
-> Rules:
 activate(n__prod(X1:S,X2:S)) -> prod(X1:S,X2:S)
 activate(n__s(X:S)) -> s(X:S)
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 add(0,X:S) -> X:S
 fact(X:S) -> if(zero(X:S),n__s(0),n__prod(X:S,fact(p(X:S))))
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 if(ttrue,X:S,Y:S) -> activate(X:S)
 p(s(X:S)) -> X:S
 prod(s(X:S),Y:S) -> add(Y:S,prod(X:S,Y:S))
 prod(0,X:S) -> 0
 prod(X1:S,X2:S) -> n__prod(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
 zero(s(X:S)) -> ffalse
 zero(0) -> ttrue

Problem 1: 

Infinite Processor:
-> Pairs:
 ACTIVATE(n__prod(X1:S,X2:S)) -> PROD(X1:S,X2:S)
 ACTIVATE(n__s(X:S)) -> S(X:S)
 ADD(s(X:S),Y:S) -> ADD(X:S,Y:S)
 ADD(s(X:S),Y:S) -> S(add(X:S,Y:S))
 FACT(X:S) -> FACT(p(X:S))
 FACT(X:S) -> IF(zero(X:S),n__s(0),n__prod(X:S,fact(p(X:S))))
 FACT(X:S) -> P(X:S)
 FACT(X:S) -> ZERO(X:S)
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
 IF(ttrue,X:S,Y:S) -> ACTIVATE(X:S)
 PROD(s(X:S),Y:S) -> ADD(Y:S,prod(X:S,Y:S))
 PROD(s(X:S),Y:S) -> PROD(X:S,Y:S)
-> Rules:
 activate(n__prod(X1:S,X2:S)) -> prod(X1:S,X2:S)
 activate(n__s(X:S)) -> s(X:S)
 activate(X:S) -> X:S
 add(s(X:S),Y:S) -> s(add(X:S,Y:S))
 add(0,X:S) -> X:S
 fact(X:S) -> if(zero(X:S),n__s(0),n__prod(X:S,fact(p(X:S))))
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 if(ttrue,X:S,Y:S) -> activate(X:S)
 p(s(X:S)) -> X:S
 prod(s(X:S),Y:S) -> add(Y:S,prod(X:S,Y:S))
 prod(0,X:S) -> 0
 prod(X1:S,X2:S) -> n__prod(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
 zero(s(X:S)) -> ffalse
 zero(0) -> ttrue
-> Pairs in cycle:
 FACT(X:S) -> FACT(p(X:S))

The problem is infinite.