YES

Problem 1: 

(VAR v_NonEmpty:S X:S Y:S)
(RULES
activate(n__f(X:S)) -> f(activate(X:S))
activate(n__true) -> ttrue
activate(X:S) -> X:S
f(X:S) -> if(X:S,c,n__f(n__true))
f(X:S) -> n__f(X:S)
if(ttrue,X:S,Y:S) -> X:S
if(ffalse,X:S,Y:S) -> activate(Y:S)
ttrue -> n__true
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__f(X:S)) -> F(activate(X:S))
 ACTIVATE(n__true) -> TRUE
 F(X:S) -> IF(X:S,c,n__f(n__true))
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__f(X:S)) -> F(activate(X:S))
 ACTIVATE(n__true) -> TRUE
 F(X:S) -> IF(X:S,c,n__f(n__true))
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__f(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__f(X:S)) -> F(activate(X:S))
 F(X:S) -> IF(X:S,c,n__f(n__true))
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
->->-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__f(X:S)) -> F(activate(X:S))
 F(X:S) -> IF(X:S,c,n__f(n__true))
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true
-> Usable rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = 2.X
[f](X) = 2.X + 2
[if](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[true] = 0
[c] = 0
[false] = 2
[n__f](X) = 2.X + 1
[n__true] = 0
[ACTIVATE](X) = 2.X
[F](X) = 2.X + 2
[IF](X1,X2,X3) = 2.X2 + 2.X3

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> F(activate(X:S))
 F(X:S) -> IF(X:S,c,n__f(n__true))
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__f(X:S)) -> F(activate(X:S))
 F(X:S) -> IF(X:S,c,n__f(n__true))
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
->->-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__f(X:S)) -> F(activate(X:S))
 F(X:S) -> IF(X:S,c,n__f(n__true))
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true
-> Usable rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[f](X) = 2.X + 2
[if](X1,X2,X3) = 2.X1 + 2.X2 + X3
[true] = 0
[c] = 0
[false] = 1
[n__f](X) = 2.X + 2
[n__true] = 0
[ACTIVATE](X) = X + 1
[F](X) = 2.X + 2
[IF](X1,X2,X3) = 2.X1 + 2.X2 + X3

Problem 1: 

SCC Processor:
-> Pairs:
 F(X:S) -> IF(X:S,c,n__f(n__true))
 IF(ffalse,X:S,Y:S) -> ACTIVATE(Y:S)
-> Rules:
 activate(n__f(X:S)) -> f(activate(X:S))
 activate(n__true) -> ttrue
 activate(X:S) -> X:S
 f(X:S) -> if(X:S,c,n__f(n__true))
 f(X:S) -> n__f(X:S)
 if(ttrue,X:S,Y:S) -> X:S
 if(ffalse,X:S,Y:S) -> activate(Y:S)
 ttrue -> n__true
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.