/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR X X1 X2 Y Z)
(RULES
activate(n__add(X1,X2)) -> add(activate(X1),X2)
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(n__from(X)) -> from(X)
activate(n__s(X)) -> s(X)
activate(X) -> X
add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
add(0,X) -> activate(X)
add(X1,X2) -> n__add(X1,X2)
and(false,Y) -> false
and(true,X) -> activate(X)
first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
first(0,X) -> nil
first(X1,X2) -> n__first(X1,X2)
from(X) -> cons(activate(X),n__from(n__s(activate(X))))
from(X) -> n__from(X)
if(false,X,Y) -> activate(Y)
if(true,X,Y) -> activate(X)
s(X) -> n__s(X)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),X2)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 ACTIVATE(n__s(X)) -> S(X)
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> ACTIVATE(Y)
 ADD(s(X),Y) -> S(n__add(activate(X),activate(Y)))
 ADD(0,X) -> ACTIVATE(X)
 AND(true,X) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
 IF(false,X,Y) -> ACTIVATE(Y)
 IF(true,X,Y) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),X2)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 ACTIVATE(n__s(X)) -> S(X)
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> ACTIVATE(Y)
 ADD(s(X),Y) -> S(n__add(activate(X),activate(Y)))
 ADD(0,X) -> ACTIVATE(X)
 AND(true,X) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
 IF(false,X,Y) -> ACTIVATE(Y)
 IF(true,X,Y) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),X2)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> ACTIVATE(Y)
 ADD(0,X) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),X2)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> ACTIVATE(Y)
 ADD(0,X) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = 2.X1 + 2.X2 + 2
[first](X1,X2) = X1 + 2.X2 + 1
[from](X) = X + 2
[s](X) = X + 1
[0] = 1
[cons](X1,X2) = 1/2.X1 + 1/2.X2 + 1/2
[n__add](X1,X2) = 2.X1 + 2.X2 + 2
[n__first](X1,X2) = X1 + 2.X2 + 1
[n__from](X) = X + 2
[n__s](X) = X + 1
[nil] = 1/2
[ACTIVATE](X) = X
[ADD](X1,X2) = X1 + X2 + 1/2
[FIRST](X1,X2) = X1 + 2.X2 + 1
[FROM](X) = X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),X2)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> ACTIVATE(Y)
 ADD(0,X) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),X2)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> ACTIVATE(Y)
 ADD(0,X) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__add(X1,X2)) -> ADD(activate(X1),X2)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> ACTIVATE(Y)
 ADD(0,X) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = 2.X1 + 2.X2 + 2
[first](X1,X2) = 2.X1 + 2.X2 + 1
[from](X) = X + 2
[s](X) = 1/2.X + 2
[0] = 0
[cons](X1,X2) = 1/2.X1 + 1/2.X2
[n__add](X1,X2) = 2.X1 + 2.X2 + 2
[n__first](X1,X2) = 2.X1 + 2.X2 + 1
[n__from](X) = X + 2
[n__s](X) = 1/2.X + 2
[nil] = 0
[ACTIVATE](X) = 1/2.X + 1/2
[ADD](X1,X2) = X1 + X2 + 1/2
[FIRST](X1,X2) = X1 + X2 + 1
[FROM](X) = 1/2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 ADD(s(X),Y) -> ACTIVATE(X)
 ADD(s(X),Y) -> ACTIVATE(Y)
 ADD(0,X) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X1)
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = X1 + 2.X2 + 1/2
[first](X1,X2) = X1 + 2.X2 + 2
[from](X) = X + 2
[s](X) = X + 2
[0] = 1/2
[cons](X1,X2) = 1/2.X1 + 1/2.X2
[n__add](X1,X2) = X1 + 2.X2 + 1/2
[n__first](X1,X2) = X1 + 2.X2 + 2
[n__from](X) = X + 2
[n__s](X) = X + 2
[nil] = 0
[ACTIVATE](X) = 1/2.X + 2
[FIRST](X1,X2) = 1/2.X1 + X2 + 2
[FROM](X) = 1/2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> ACTIVATE(X2)
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = 1/2.X1 + X2 + 1
[first](X1,X2) = 2.X1 + 2.X2 + 1/2
[from](X) = 2.X + 2
[s](X) = 1/2.X
[0] = 2
[cons](X1,X2) = X1 + 1/2.X2
[n__add](X1,X2) = 1/2.X1 + X2 + 1
[n__first](X1,X2) = 2.X1 + 2.X2 + 1/2
[n__from](X) = 2.X + 2
[n__s](X) = 1/2.X
[nil] = 0
[ACTIVATE](X) = 1/2.X
[FIRST](X1,X2) = X1 + X2
[FROM](X) = X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__first(X1,X2)) -> FIRST(activate(X1),activate(X2))
 ACTIVATE(n__from(X)) -> FROM(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
-> Usable rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 s(X) -> n__s(X)
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[activate](X) = X
[add](X1,X2) = 2.X2 + 1/2
[first](X1,X2) = X1 + 2.X2 + 1
[from](X) = 2.X
[s](X) = X
[0] = 1
[cons](X1,X2) = 1/2.X1 + 1/2.X2
[n__add](X1,X2) = 2.X2 + 1/2
[n__first](X1,X2) = X1 + 2.X2 + 1
[n__from](X) = 2.X
[n__s](X) = X
[nil] = 1
[ACTIVATE](X) = X
[FIRST](X1,X2) = X1 + 2.X2
[FROM](X) = X

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__from(X)) -> FROM(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(X)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Y)
 FIRST(s(X),cons(Y,Z)) -> ACTIVATE(Z)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__from(X)) -> FROM(X)
 FROM(X) -> ACTIVATE(X)
->->-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)

Problem 1: 

Subterm Processor:
-> Pairs:
 ACTIVATE(n__from(X)) -> FROM(X)
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
->Projection:
 pi(ACTIVATE) = 1
 pi(FROM) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 FROM(X) -> ACTIVATE(X)
-> Rules:
 activate(n__add(X1,X2)) -> add(activate(X1),X2)
 activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
 activate(n__from(X)) -> from(X)
 activate(n__s(X)) -> s(X)
 activate(X) -> X
 add(s(X),Y) -> s(n__add(activate(X),activate(Y)))
 add(0,X) -> activate(X)
 add(X1,X2) -> n__add(X1,X2)
 and(false,Y) -> false
 and(true,X) -> activate(X)
 first(s(X),cons(Y,Z)) -> cons(activate(Y),n__first(activate(X),activate(Z)))
 first(0,X) -> nil
 first(X1,X2) -> n__first(X1,X2)
 from(X) -> cons(activate(X),n__from(n__s(activate(X))))
 from(X) -> n__from(X)
 if(false,X,Y) -> activate(Y)
 if(true,X,Y) -> activate(X)
 s(X) -> n__s(X)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.