YES

Problem 1: 

(VAR v_NonEmpty:S X:S X1:S X2:S XS:S)
(RULES
active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
active(tail(cons(X:S,XS:S))) -> mark(XS:S)
active(tail(X:S)) -> tail(active(X:S))
active(zeros) -> mark(cons(0,zeros))
cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
proper(tail(X:S)) -> tail(proper(X:S))
proper(0) -> ok(0)
proper(zeros) -> ok(zeros)
tail(mark(X:S)) -> mark(tail(X:S))
tail(ok(X:S)) -> ok(tail(X:S))
top(mark(X:S)) -> top(proper(X:S))
top(ok(X:S)) -> top(active(X:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S)
 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
 ACTIVE(tail(X:S)) -> TAIL(active(X:S))
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S))
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(tail(X:S)) -> PROPER(X:S)
 PROPER(tail(X:S)) -> TAIL(proper(X:S))
 TAIL(mark(X:S)) -> TAIL(X:S)
 TAIL(ok(X:S)) -> TAIL(X:S)
 TOP(mark(X:S)) -> PROPER(X:S)
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> ACTIVE(X:S)
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(cons(X1:S,X2:S)) -> CONS(active(X1:S),X2:S)
 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
 ACTIVE(tail(X:S)) -> TAIL(active(X:S))
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
 PROPER(cons(X1:S,X2:S)) -> CONS(proper(X1:S),proper(X2:S))
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(tail(X:S)) -> PROPER(X:S)
 PROPER(tail(X:S)) -> TAIL(proper(X:S))
 TAIL(mark(X:S)) -> TAIL(X:S)
 TAIL(ok(X:S)) -> TAIL(X:S)
 TOP(mark(X:S)) -> PROPER(X:S)
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> ACTIVE(X:S)
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TAIL(mark(X:S)) -> TAIL(X:S)
 TAIL(ok(X:S)) -> TAIL(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(tail(X:S)) -> PROPER(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->->Cycle:
->->-> Pairs:
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> TOP(active(X:S))
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))


The problem is decomposed in 5 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 TAIL(mark(X:S)) -> TAIL(X:S)
 TAIL(ok(X:S)) -> TAIL(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(TAIL) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 CONS(mark(X1:S),X2:S) -> CONS(X1:S,X2:S)
 CONS(ok(X1:S),ok(X2:S)) -> CONS(X1:S,X2:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(CONS) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 PROPER(cons(X1:S,X2:S)) -> PROPER(X1:S)
 PROPER(cons(X1:S,X2:S)) -> PROPER(X2:S)
 PROPER(tail(X:S)) -> PROPER(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(PROPER) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 ACTIVE(cons(X1:S,X2:S)) -> ACTIVE(X1:S)
 ACTIVE(tail(X:S)) -> ACTIVE(X:S)
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Projection:
 pi(ACTIVE) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Reduction Pairs Processor:
-> Pairs:
 TOP(mark(X:S)) -> TOP(proper(X:S))
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
-> Usable rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[cons](X1,X2) = X1 + 1/2.X2
[proper](X) = X
[tail](X) = 2.X + 1/2
[top](X) = 0
[0] = 1/2
[fSNonEmpty] = 0
[mark](X) = X + 1/2
[ok](X) = X
[zeros] = 2
[ACTIVE](X) = 0
[CONS](X1,X2) = 0
[PROPER](X) = 0
[TAIL](X) = 0
[TOP](X) = 1/2.X

Problem 1.5: 

SCC Processor:
-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
->->-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))

Problem 1.5: 

Reduction Pairs Processor:
-> Pairs:
 TOP(ok(X:S)) -> TOP(active(X:S))
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
-> Usable rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = 2.X
[cons](X1,X2) = X1
[proper](X) = 0
[tail](X) = 2.X + 1
[top](X) = 0
[0] = 2
[fSNonEmpty] = 0
[mark](X) = 0
[ok](X) = 2.X + 2
[zeros] = 0
[ACTIVE](X) = 0
[CONS](X1,X2) = 0
[PROPER](X) = 0
[TAIL](X) = 0
[TOP](X) = X

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(cons(X1:S,X2:S)) -> cons(active(X1:S),X2:S)
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(tail(X:S)) -> tail(active(X:S))
 active(zeros) -> mark(cons(0,zeros))
 cons(mark(X1:S),X2:S) -> mark(cons(X1:S,X2:S))
 cons(ok(X1:S),ok(X2:S)) -> ok(cons(X1:S,X2:S))
 proper(cons(X1:S,X2:S)) -> cons(proper(X1:S),proper(X2:S))
 proper(tail(X:S)) -> tail(proper(X:S))
 proper(0) -> ok(0)
 proper(zeros) -> ok(zeros)
 tail(mark(X:S)) -> mark(tail(X:S))
 tail(ok(X:S)) -> ok(tail(X:S))
 top(mark(X:S)) -> top(proper(X:S))
 top(ok(X:S)) -> top(active(X:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.