YES

Problem 1: 

(VAR v_NonEmpty:S L:S X:S X1:S X2:S)
(RULES
active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
active(adx(nil)) -> mark(nil)
active(head(cons(X:S,L:S))) -> mark(X:S)
active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
active(incr(nil)) -> mark(nil)
active(tail(cons(X:S,L:S))) -> mark(L: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)
head(active(X:S)) -> head(X:S)
head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
mark(head(X:S)) -> active(head(mark(X:S)))
mark(incr(X:S)) -> active(incr(mark(X:S)))
mark(s(X:S)) -> active(s(mark(X:S)))
mark(tail(X:S)) -> active(tail(mark(X:S)))
mark(0) -> active(0)
mark(nats) -> active(nats)
mark(nil) -> active(nil)
mark(zeros) -> active(zeros)
s(active(X:S)) -> s(X:S)
s(mark(X:S)) -> s(X:S)
tail(active(X:S)) -> tail(X:S)
tail(mark(X:S)) -> tail(X:S)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,L:S))) -> ADX(L:S)
 ACTIVE(adx(cons(X:S,L:S))) -> CONS(X:S,adx(L:S))
 ACTIVE(adx(cons(X:S,L:S))) -> INCR(cons(X:S,adx(L:S)))
 ACTIVE(adx(cons(X:S,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(adx(nil)) -> MARK(nil)
 ACTIVE(head(cons(X:S,L:S))) -> MARK(X:S)
 ACTIVE(incr(cons(X:S,L:S))) -> CONS(s(X:S),incr(L:S))
 ACTIVE(incr(cons(X:S,L:S))) -> INCR(L:S)
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(incr(cons(X:S,L:S))) -> S(X:S)
 ACTIVE(incr(nil)) -> MARK(nil)
 ACTIVE(tail(cons(X:S,L:S))) -> MARK(L: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)
 HEAD(active(X:S)) -> HEAD(X:S)
 HEAD(mark(X:S)) -> HEAD(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(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(head(X:S)) -> ACTIVE(head(mark(X:S)))
 MARK(head(X:S)) -> HEAD(mark(X:S))
 MARK(head(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(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> S(mark(X:S))
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> TAIL(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)
 TAIL(active(X:S)) -> TAIL(X:S)
 TAIL(mark(X:S)) -> TAIL(X:S)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,L:S))) -> ADX(L:S)
 ACTIVE(adx(cons(X:S,L:S))) -> CONS(X:S,adx(L:S))
 ACTIVE(adx(cons(X:S,L:S))) -> INCR(cons(X:S,adx(L:S)))
 ACTIVE(adx(cons(X:S,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(adx(nil)) -> MARK(nil)
 ACTIVE(head(cons(X:S,L:S))) -> MARK(X:S)
 ACTIVE(incr(cons(X:S,L:S))) -> CONS(s(X:S),incr(L:S))
 ACTIVE(incr(cons(X:S,L:S))) -> INCR(L:S)
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(incr(cons(X:S,L:S))) -> S(X:S)
 ACTIVE(incr(nil)) -> MARK(nil)
 ACTIVE(tail(cons(X:S,L:S))) -> MARK(L: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)
 HEAD(active(X:S)) -> HEAD(X:S)
 HEAD(mark(X:S)) -> HEAD(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(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> CONS(mark(X1:S),X2:S)
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(head(X:S)) -> ACTIVE(head(mark(X:S)))
 MARK(head(X:S)) -> HEAD(mark(X:S))
 MARK(head(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(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> S(mark(X:S))
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> TAIL(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)
 TAIL(active(X:S)) -> TAIL(X:S)
 TAIL(mark(X:S)) -> TAIL(X:S)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TAIL(active(X:S)) -> TAIL(X:S)
 TAIL(mark(X:S)) -> TAIL(X:S)
->->-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->->Cycle:
->->-> Pairs:
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
->->-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->->Cycle:
->->-> Pairs:
 INCR(active(X:S)) -> INCR(X:S)
 INCR(mark(X:S)) -> INCR(X:S)
->->-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->->Cycle:
->->-> Pairs:
 HEAD(active(X:S)) -> HEAD(X:S)
 HEAD(mark(X:S)) -> HEAD(X:S)
->->-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->->Cycle:
->->-> Pairs:
 ADX(active(X:S)) -> ADX(X:S)
 ADX(mark(X:S)) -> ADX(X:S)
->->-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->->Cycle:
->->-> Pairs:
 ACTIVE(adx(cons(X:S,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(head(cons(X:S,L:S))) -> MARK(X:S)
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(tail(cons(X:S,L:S))) -> MARK(L:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(head(X:S)) -> ACTIVE(head(mark(X:S)))
 MARK(head(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
 MARK(zeros) -> ACTIVE(zeros)
->->-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)


The problem is decomposed in 7 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 TAIL(active(X:S)) -> TAIL(X:S)
 TAIL(mark(X:S)) -> TAIL(X:S)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->Projection:
 pi(TAIL) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->Projection:
 pi(S) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->Projection:
 pi(INCR) = 1

Problem 1.3: 

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

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 HEAD(active(X:S)) -> HEAD(X:S)
 HEAD(mark(X:S)) -> HEAD(X:S)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->Projection:
 pi(HEAD) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->Projection:
 pi(CONS) = 2

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->Projection:
 pi(ADX) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(head(cons(X:S,L:S))) -> MARK(X:S)
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(tail(cons(X:S,L:S))) -> MARK(L:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(head(X:S)) -> ACTIVE(head(mark(X:S)))
 MARK(head(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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
[head](X) = 2.X + 2
[incr](X) = X
[mark](X) = X
[s](X) = X
[tail](X) = 2.X + 1
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[nil] = 2
[zeros] = 0
[ACTIVE](X) = 2.X + 1
[ADX](X) = 0
[CONS](X1,X2) = 0
[HEAD](X) = 0
[INCR](X) = 0
[MARK](X) = 2.X + 1
[S](X) = 0
[TAIL](X) = 0

Problem 1.7: 

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

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(tail(cons(X:S,L:S))) -> MARK(L:S)
 ACTIVE(nats) -> MARK(adx(zeros))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(head(X:S)) -> ACTIVE(head(mark(X:S)))
 MARK(head(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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
[head](X) = 2.X + 2
[incr](X) = X
[mark](X) = X
[s](X) = X
[tail](X) = X + 2
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[nil] = 2
[zeros] = 0
[ACTIVE](X) = 2.X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HEAD](X) = 0
[INCR](X) = 0
[MARK](X) = 2.X + 2
[S](X) = 0
[TAIL](X) = 0

Problem 1.7: 

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

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(nats) -> MARK(adx(zeros))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(adx(X:S)) -> MARK(X:S)
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(head(X:S)) -> ACTIVE(head(mark(X:S)))
 MARK(head(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(nats) -> ACTIVE(nats)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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
[head](X) = 2.X + 1
[incr](X) = X
[mark](X) = X
[s](X) = X
[tail](X) = X + 1
[0] = 0
[fSNonEmpty] = 0
[nats] = 1
[nil] = 2
[zeros] = 0
[ACTIVE](X) = 2.X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HEAD](X) = 0
[INCR](X) = 0
[MARK](X) = 2.X + 2
[S](X) = 0
[TAIL](X) = 0

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(head(X:S)) -> ACTIVE(head(mark(X:S)))
 MARK(head(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = 0
[adx](X) = 1
[cons](X1,X2) = 0
[head](X) = 0
[incr](X) = 1
[mark](X) = 0
[s](X) = 1
[tail](X) = 0
[0] = 2
[fSNonEmpty] = 0
[nats] = 0
[nil] = 2
[zeros] = 1
[ACTIVE](X) = X + 1
[ADX](X) = 0
[CONS](X1,X2) = 0
[HEAD](X) = 0
[INCR](X) = 0
[MARK](X) = 2
[S](X) = 0
[TAIL](X) = 0

Problem 1.7: 

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

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(head(X:S)) -> MARK(X:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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
[head](X) = 2.X + 1
[incr](X) = X
[mark](X) = X
[s](X) = X
[tail](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[nil] = 2
[zeros] = 0
[ACTIVE](X) = 2.X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HEAD](X) = 0
[INCR](X) = 0
[MARK](X) = 2.X + 2
[S](X) = 0
[TAIL](X) = 0

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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
[head](X) = 2.X + 1
[incr](X) = X
[mark](X) = X
[s](X) = X
[tail](X) = 2.X + 1
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[nil] = 2
[zeros] = 0
[ACTIVE](X) = 2.X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HEAD](X) = 0
[INCR](X) = 0
[MARK](X) = 2.X + 2
[S](X) = 0
[TAIL](X) = 0

Problem 1.7: 

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

Problem 1.7: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(adx(cons(X:S,L:S))) -> MARK(incr(cons(X:S,adx(L:S))))
 ACTIVE(incr(cons(X:S,L:S))) -> MARK(cons(s(X:S),incr(L:S)))
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(adx(X:S)) -> ACTIVE(adx(mark(X:S)))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(incr(X:S)) -> ACTIVE(incr(mark(X:S)))
 MARK(incr(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(adx(cons(X:S,L:S))) -> mark(incr(cons(X:S,adx(L:S))))
 active(adx(nil)) -> mark(nil)
 active(head(cons(X:S,L:S))) -> mark(X:S)
 active(incr(cons(X:S,L:S))) -> mark(cons(s(X:S),incr(L:S)))
 active(incr(nil)) -> mark(nil)
 active(tail(cons(X:S,L:S))) -> mark(L: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)
 head(active(X:S)) -> head(X:S)
 head(mark(X:S)) -> head(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(mark(X1:S),X2:S))
 mark(head(X:S)) -> active(head(mark(X:S)))
 mark(incr(X:S)) -> active(incr(mark(X:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(nats) -> active(nats)
 mark(nil) -> active(nil)
 mark(zeros) -> active(zeros)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(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
[head](X) = 2.X + 2
[incr](X) = X + 1/2
[mark](X) = X
[s](X) = X
[tail](X) = 2.X
[0] = 0
[fSNonEmpty] = 0
[nats] = 2
[nil] = 1/2
[zeros] = 0
[ACTIVE](X) = 1/2.X + 2
[ADX](X) = 0
[CONS](X1,X2) = 0
[HEAD](X) = 0
[INCR](X) = 0
[MARK](X) = 1/2.X + 2
[S](X) = 0
[TAIL](X) = 0

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

Problem 1.7: 

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

The problem is finite.