YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S)
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 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)
 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(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> TAIL(mark(X:S))
 MARK(zeros) -> ACTIVE(zeros)
 TAIL(active(X:S)) -> TAIL(X:S)
 TAIL(mark(X:S)) -> TAIL(X:S)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S)
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 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)
 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(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(tail(X:S)) -> TAIL(mark(X:S))
 MARK(zeros) -> ACTIVE(zeros)
 TAIL(active(X:S)) -> TAIL(X:S)
 TAIL(mark(X:S)) -> TAIL(X:S)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->->Cycle:
->->-> Pairs:
 ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S)
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
->->-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.2: 

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(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)

Problem 1.2: 

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(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
->Projection:
 pi(CONS) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(tail(cons(X:S,XS:S))) -> MARK(XS:S)
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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
[cons](X1,X2) = 2.X1 + X2
[mark](X) = 2.X
[tail](X) = 2.X + 1
[0] = 0
[fSNonEmpty] = 0
[zeros] = 0
[ACTIVE](X) = X + 2
[CONS](X1,X2) = 0
[MARK](X) = 2.X + 2
[TAIL](X) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
->->-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1:S,X2:S)) -> ACTIVE(cons(mark(X1:S),X2:S))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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
[cons](X1,X2) = 1
[mark](X) = 2
[tail](X) = 2
[0] = 2
[fSNonEmpty] = 0
[zeros] = 2
[ACTIVE](X) = X
[CONS](X1,X2) = 0
[MARK](X) = 2
[TAIL](X) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
->->-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> ACTIVE(tail(mark(X:S)))
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)
-> Usable rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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
[cons](X1,X2) = X1 + X2
[mark](X) = X
[tail](X) = 2.X + 2
[0] = 0
[fSNonEmpty] = 0
[zeros] = 1
[ACTIVE](X) = X + 2
[CONS](X1,X2) = 0
[MARK](X) = 2.X + 1
[TAIL](X) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
->->-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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
[cons](X1,X2) = 2.X1 + 1
[mark](X) = 2.X
[tail](X) = 2.X
[0] = 0
[fSNonEmpty] = 0
[zeros] = 2
[ACTIVE](X) = 2.X + 2
[CONS](X1,X2) = 0
[MARK](X) = 2.X + 2
[TAIL](X) = 0

Problem 1.3: 

SCC Processor:
-> Pairs:
 MARK(cons(X1:S,X2:S)) -> MARK(X1:S)
 MARK(tail(X:S)) -> MARK(X:S)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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(tail(X:S)) -> MARK(X:S)
->->-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X:S)) -> tail(X:S)
 tail(mark(X:S)) -> tail(X:S)

Problem 1.3: 

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

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 active(tail(cons(X:S,XS:S))) -> mark(XS:S)
 active(zeros) -> mark(cons(0,zeros))
 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)
 mark(cons(X1:S,X2:S)) -> active(cons(mark(X1:S),X2:S))
 mark(tail(X:S)) -> active(tail(mark(X:S)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 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.