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


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YES

Problem 1: 

(VAR X X1 X2 XS)
(RULES
active(tail(cons(X,XS))) -> mark(XS)
active(zeros) -> mark(cons(0,zeros))
cons(active(X1),X2) -> cons(X1,X2)
cons(mark(X1),X2) -> cons(X1,X2)
cons(X1,active(X2)) -> cons(X1,X2)
cons(X1,mark(X2)) -> cons(X1,X2)
mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
mark(tail(X)) -> active(tail(mark(X)))
mark(0) -> active(0)
mark(zeros) -> active(zeros)
tail(active(X)) -> tail(X)
tail(mark(X)) -> tail(X)
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(tail(cons(X,XS))) -> MARK(XS)
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2))
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(tail(X)) -> ACTIVE(tail(mark(X)))
 MARK(tail(X)) -> MARK(X)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X,XS))) -> mark(XS)
 active(zeros) -> mark(cons(0,zeros))
 cons(active(X1),X2) -> cons(X1,X2)
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
 mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
 mark(tail(X)) -> active(tail(mark(X)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X)) -> tail(X)
 tail(mark(X)) -> tail(X)
-> Usable rules:
 active(tail(cons(X,XS))) -> mark(XS)
 active(zeros) -> mark(cons(0,zeros))
 cons(active(X1),X2) -> cons(X1,X2)
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
 mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
 mark(tail(X)) -> active(tail(mark(X)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X)) -> tail(X)
 tail(mark(X)) -> tail(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[cons](X1,X2) = 2.X1 + 2.X2
[mark](X) = 2.X
[tail](X) = 2.X + 1
[0] = 0
[zeros] = 0
[ACTIVE](X) = X
[MARK](X) = 2.X

Problem 1.3: 

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

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1,X2)) -> ACTIVE(cons(mark(X1),X2))
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(tail(X)) -> ACTIVE(tail(mark(X)))
 MARK(tail(X)) -> MARK(X)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X,XS))) -> mark(XS)
 active(zeros) -> mark(cons(0,zeros))
 cons(active(X1),X2) -> cons(X1,X2)
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
 mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
 mark(tail(X)) -> active(tail(mark(X)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X)) -> tail(X)
 tail(mark(X)) -> tail(X)
-> Usable rules:
 active(tail(cons(X,XS))) -> mark(XS)
 active(zeros) -> mark(cons(0,zeros))
 cons(active(X1),X2) -> cons(X1,X2)
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
 mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
 mark(tail(X)) -> active(tail(mark(X)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X)) -> tail(X)
 tail(mark(X)) -> tail(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = 2
[cons](X1,X2) = 0
[mark](X) = 2
[tail](X) = 2
[0] = 0
[zeros] = 2
[ACTIVE](X) = X
[MARK](X) = 2

Problem 1.3: 

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

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(tail(X)) -> ACTIVE(tail(mark(X)))
 MARK(tail(X)) -> MARK(X)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X,XS))) -> mark(XS)
 active(zeros) -> mark(cons(0,zeros))
 cons(active(X1),X2) -> cons(X1,X2)
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
 mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
 mark(tail(X)) -> active(tail(mark(X)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X)) -> tail(X)
 tail(mark(X)) -> tail(X)
-> Usable rules:
 active(tail(cons(X,XS))) -> mark(XS)
 active(zeros) -> mark(cons(0,zeros))
 cons(active(X1),X2) -> cons(X1,X2)
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
 mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
 mark(tail(X)) -> active(tail(mark(X)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X)) -> tail(X)
 tail(mark(X)) -> tail(X)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = X
[cons](X1,X2) = 2.X1 + 2.X2
[mark](X) = 2.X
[tail](X) = 2.X + 2
[0] = 0
[zeros] = 0
[ACTIVE](X) = 2
[MARK](X) = 2.X + 2

Problem 1.3: 

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

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 ACTIVE(zeros) -> MARK(cons(0,zeros))
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(tail(X)) -> MARK(X)
 MARK(zeros) -> ACTIVE(zeros)
-> Rules:
 active(tail(cons(X,XS))) -> mark(XS)
 active(zeros) -> mark(cons(0,zeros))
 cons(active(X1),X2) -> cons(X1,X2)
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
 mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
 mark(tail(X)) -> active(tail(mark(X)))
 mark(0) -> active(0)
 mark(zeros) -> active(zeros)
 tail(active(X)) -> tail(X)
 tail(mark(X)) -> tail(X)
-> Usable rules:
 cons(active(X1),X2) -> cons(X1,X2)
 cons(mark(X1),X2) -> cons(X1,X2)
 cons(X1,active(X2)) -> cons(X1,X2)
 cons(X1,mark(X2)) -> cons(X1,X2)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[active](X) = 2.X
[cons](X1,X2) = 2.X1 + 1
[mark](X) = X
[tail](X) = 2.X
[0] = 0
[zeros] = 2
[ACTIVE](X) = X + 2
[MARK](X) = 2.X + 1

Problem 1.3: 

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

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.