YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S Y:S)
(RULES
active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
active(hd(cons(X:S,Y:S))) -> mark(X:S)
active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
active(tl(cons(X:S,Y:S))) -> mark(Y:S)
active(nats) -> mark(adx(zeros))
active(zeros) -> mark(cons(0,zeros))
adx(active(X:S)) -> adx(X:S)
adx(mark(X:S)) -> adx(X:S)
cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
hd(active(X:S)) -> hd(X:S)
hd(mark(X:S)) -> hd(X:S)
incr(active(X:S)) -> incr(X:S)
incr(mark(X:S)) -> incr(X:S)
mark(adx(X:S)) -> active(adx(mark(X:S)))
mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
mark(hd(X:S)) -> active(hd(mark(X:S)))
mark(incr(X:S)) -> active(incr(mark(X:S)))
mark(s(X:S)) -> active(s(X:S))
mark(tl(X:S)) -> active(tl(mark(X:S)))
mark(0) -> active(0)
mark(nats) -> active(nats)
mark(zeros) -> active(zeros)
s(active(X:S)) -> s(X:S)
s(mark(X:S)) -> s(X:S)
tl(active(X:S)) -> tl(X:S)
tl(mark(X:S)) -> tl(X:S)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> ADX(Y:S)
 ACTIVE(adx(cons(X:S,Y:S))) -> CONS(X:S,adx(Y:S))
 ACTIVE(adx(cons(X:S,Y:S))) -> INCR(cons(X:S,adx(Y:S)))
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(hd(cons(X:S,Y:S))) -> MARK(X:S)
 ACTIVE(incr(cons(X:S,Y:S))) -> CONS(s(X:S),incr(Y:S))
 ACTIVE(incr(cons(X:S,Y:S))) -> INCR(Y:S)
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(incr(cons(X:S,Y:S))) -> S(X:S)
 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 ADX(active(X:S)) -> ADX(X:S)
 ADX(mark(X:S)) -> ADX(X:S)
 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
 HD(active(X:S)) -> HD(X:S)
 HD(mark(X:S)) -> HD(X:S)
 INCR(active(X:S)) -> INCR(X:S)
 INCR(mark(X:S)) -> INCR(X:S)
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> ADX(mark(X:S))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(X1:S,X2:S))
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> HD(mark(X:S))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> INCR(mark(X:S))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> ACTIVE(s(X:S))
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> TL(mark(X:S))
 MARK(nats) -> ACTIVE(nats)
 MARK(zeros) -> ACTIVE(zeros)
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
 TL(active(X:S)) -> TL(X:S)
 TL(mark(X:S)) -> TL(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> ADX(Y:S)
 ACTIVE(adx(cons(X:S,Y:S))) -> CONS(X:S,adx(Y:S))
 ACTIVE(adx(cons(X:S,Y:S))) -> INCR(cons(X:S,adx(Y:S)))
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(hd(cons(X:S,Y:S))) -> MARK(X:S)
 ACTIVE(incr(cons(X:S,Y:S))) -> CONS(s(X:S),incr(Y:S))
 ACTIVE(incr(cons(X:S,Y:S))) -> INCR(Y:S)
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(incr(cons(X:S,Y:S))) -> S(X:S)
 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 ADX(active(X:S)) -> ADX(X:S)
 ADX(mark(X:S)) -> ADX(X:S)
 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
 HD(active(X:S)) -> HD(X:S)
 HD(mark(X:S)) -> HD(X:S)
 INCR(active(X:S)) -> INCR(X:S)
 INCR(mark(X:S)) -> INCR(X:S)
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> ADX(mark(X:S))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(X1:S,X2:S))
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> HD(mark(X:S))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> INCR(mark(X:S))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> ACTIVE(s(X:S))
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> TL(mark(X:S))
 MARK(nats) -> ACTIVE(nats)
 MARK(zeros) -> ACTIVE(zeros)
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
 TL(active(X:S)) -> TL(X:S)
 TL(mark(X:S)) -> TL(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TL(active(X:S)) -> TL(X:S)
 TL(mark(X:S)) -> TL(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->->Cycle:
->->-> Pairs:
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->->Cycle:
->->-> Pairs:
 INCR(active(X:S)) -> INCR(X:S)
 INCR(mark(X:S)) -> INCR(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->->Cycle:
->->-> Pairs:
 HD(active(X:S)) -> HD(X:S)
 HD(mark(X:S)) -> HD(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->->Cycle:
->->-> Pairs:
 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->->Cycle:
->->-> Pairs:
 ADX(active(X:S)) -> ADX(X:S)
 ADX(mark(X:S)) -> ADX(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(hd(cons(X:S,Y:S))) -> MARK(X:S)
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)


The problem is decomposed in 7 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 TL(active(X:S)) -> TL(X:S)
 TL(mark(X:S)) -> TL(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Projection:
 pi(TL) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Projection:
 pi(S) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 INCR(active(X:S)) -> INCR(X:S)
 INCR(mark(X:S)) -> INCR(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Projection:
 pi(INCR) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 HD(active(X:S)) -> HD(X:S)
 HD(mark(X:S)) -> HD(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Projection:
 pi(HD) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 CONS(active(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Projection:
 pi(CONS) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.5: 

Subterm Processor:
-> Pairs:
 CONS(X1:S,active(X2:S)) -> CONS(X1:S,X2:S)
 CONS(X1:S,mark(X2:S)) -> CONS(X1:S,X2:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Projection:
 pi(CONS) = 2

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Subterm Processor:
-> Pairs:
 ADX(active(X:S)) -> ADX(X:S)
 ADX(mark(X:S)) -> ADX(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Projection:
 pi(ADX) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(hd(cons(X:S,Y:S))) -> MARK(X:S)
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[adx](X) = X
[cons](X1,X2) = 2.X1 + 2.X2
[hd](X) = 2.X + 2
[incr](X) = X
[mark](X) = X
[s](X) = X
[tl](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 0
[ACTIVE](X) = 2.X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = 2.X + 2
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(tl(cons(X:S,Y:S))) -> MARK(Y:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[adx](X) = 2.X
[cons](X1,X2) = 2.X1 + 2.X2
[hd](X) = X
[incr](X) = X
[mark](X) = X
[s](X) = 0
[tl](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 0
[ACTIVE](X) = X
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = X
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(nats) -> MARK(adx(zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(nats) -> MARK(adx(zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 ACTIVE(nats) -> MARK(adx(zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[adx](X) = 2.X + 1
[cons](X1,X2) = 2.X1 + X2
[hd](X) = 2.X
[incr](X) = X
[mark](X) = X
[s](X) = X
[tl](X) = 2.X
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 0
[ACTIVE](X) = 2.X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = 2.X + 2
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[adx](X) = 2.X + 2
[cons](X1,X2) = X1 + X2
[hd](X) = 2.X + 1
[incr](X) = X
[mark](X) = X
[s](X) = X
[tl](X) = X + 2
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 0
[ACTIVE](X) = 2.X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = 2.X + 2
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(hd(X:S)) -> ACTIVE(hd(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = 0
[adx](X) = 1
[cons](X1,X2) = 1
[hd](X) = 0
[incr](X) = 1
[mark](X) = 0
[s](X) = 2
[tl](X) = 1
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 0
[ACTIVE](X) = 2.X
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = 2
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(hd(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[adx](X) = 2.X + 2
[cons](X1,X2) = 2.X1 + X2
[hd](X) = X + 2
[incr](X) = X
[mark](X) = X
[s](X) = X
[tl](X) = 2.X + 1
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 0
[ACTIVE](X) = 2.X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = 2.X + 2
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> ACTIVE(tl(mark(X:S)))
 MARK(tl(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = 2
[adx](X) = 2
[cons](X1,X2) = 0
[hd](X) = 2
[incr](X) = 2
[mark](X) = 2
[s](X) = 1
[tl](X) = 1
[0] = 1
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 2
[ACTIVE](X) = X
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = 2
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> MARK(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(tl(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[adx](X) = 2.X
[cons](X1,X2) = 2.X1 + 2.X2
[hd](X) = 2.X + 1
[incr](X) = X
[mark](X) = X
[s](X) = X
[tl](X) = X + 2
[0] = 0
[fSNonEmpty] = 0
[nats] = 1
[zeros] = 0
[ACTIVE](X) = X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = X + 2
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,Y:S))) -> MARK(incr(cons(X:S,adx(Y:S))))
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[adx](X) = X + 1
[cons](X1,X2) = 2.X1 + 1/2.X2 + 1/2
[hd](X) = 1/2.X
[incr](X) = X
[mark](X) = X
[s](X) = 1/2.X
[tl](X) = 2.X
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 1
[ACTIVE](X) = X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = X + 2
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = 2
[adx](X) = 0
[cons](X1,X2) = 0
[hd](X) = 2
[incr](X) = 2
[mark](X) = 2
[s](X) = 2
[tl](X) = 2
[0] = 2
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 2
[ACTIVE](X) = X
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = 2
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(incr(cons(X:S,Y:S))) -> MARK(cons(s(X:S),incr(Y:S)))
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
-> Usable rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[adx](X) = 2.X + 2
[cons](X1,X2) = 2.X1 + 1/2.X2
[hd](X) = X + 2
[incr](X) = X + 1
[mark](X) = X
[s](X) = 1/2.X
[tl](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[zeros] = 0
[ACTIVE](X) = X + 1
[ADX](X) = 0
[CONS](X1,X2) = 0
[HD](X) = 0
[INCR](X) = 0
[MARK](X) = X + 1
[S](X) = 0
[TL](X) = 0

Problem 1.7: 

SCC Processor:
-> Pairs:
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(incr(X:S)) -> MARK(X:S)
->->-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)

Problem 1.7: 

Subterm Processor:
-> Pairs:
 MARK(incr(X:S)) -> MARK(X:S)
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Projection:
 pi(MARK) = 1

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,Y:S))) -> mark(incr(cons(X:S,adx(Y:S))))
 active(hd(cons(X:S,Y:S))) -> mark(X:S)
 active(incr(cons(X:S,Y:S))) -> mark(cons(s(X:S),incr(Y:S)))
 active(tl(cons(X:S,Y:S))) -> mark(Y:S)
 active(nats) -> mark(adx(zeros))
 active(zeros) -> mark(cons(0,zeros))
 adx(active(X:S)) -> adx(X:S)
 adx(mark(X:S)) -> adx(X:S)
 cons(active(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(mark(X1:S),X2:S) -> cons(X1:S,X2:S)
 cons(X1:S,active(X2:S)) -> cons(X1:S,X2:S)
 cons(X1:S,mark(X2:S)) -> cons(X1:S,X2:S)
 hd(active(X:S)) -> hd(X:S)
 hd(mark(X:S)) -> hd(X:S)
 incr(active(X:S)) -> incr(X:S)
 incr(mark(X:S)) -> incr(X:S)
 mark(adx(X:S)) -> active(adx(mark(X:S)))
 mark(cons(X1:S,X2:S)) -> active(cons(X1:S,X2:S))
 mark(hd(X:S)) -> active(hd(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(X:S))
 mark(tl(X:S)) -> active(tl(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tl(active(X:S)) -> tl(X:S)
 tl(mark(X:S)) -> tl(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.