/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 Y)
(RULES
a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
a__adx(X) -> adx(X)
a__hd(cons(X,Y)) -> mark(X)
a__hd(X) -> hd(X)
a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
a__incr(X) -> incr(X)
a__nats -> a__adx(a__zeros)
a__nats -> nats
a__tl(cons(X,Y)) -> mark(Y)
a__tl(X) -> tl(X)
a__zeros -> cons(0,zeros)
a__zeros -> zeros
mark(0) -> 0
mark(adx(X)) -> a__adx(mark(X))
mark(cons(X1,X2)) -> cons(X1,X2)
mark(hd(X)) -> a__hd(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats) -> a__nats
mark(s(X)) -> s(X)
mark(tl(X)) -> a__tl(mark(X))
mark(zeros) -> a__zeros
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__ADX(cons(X,Y)) -> A__INCR(cons(X,adx(Y)))
 A__HD(cons(X,Y)) -> MARK(X)
 A__NATS -> A__ADX(a__zeros)
 A__NATS -> A__ZEROS
 A__TL(cons(X,Y)) -> MARK(Y)
 MARK(adx(X)) -> A__ADX(mark(X))
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> A__HD(mark(X))
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(nats) -> A__NATS
 MARK(tl(X)) -> A__TL(mark(X))
 MARK(tl(X)) -> MARK(X)
 MARK(zeros) -> A__ZEROS
-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros

Problem 1: 

SCC Processor:
-> Pairs:
 A__ADX(cons(X,Y)) -> A__INCR(cons(X,adx(Y)))
 A__HD(cons(X,Y)) -> MARK(X)
 A__NATS -> A__ADX(a__zeros)
 A__NATS -> A__ZEROS
 A__TL(cons(X,Y)) -> MARK(Y)
 MARK(adx(X)) -> A__ADX(mark(X))
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> A__HD(mark(X))
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(nats) -> A__NATS
 MARK(tl(X)) -> A__TL(mark(X))
 MARK(tl(X)) -> MARK(X)
 MARK(zeros) -> A__ZEROS
-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__HD(cons(X,Y)) -> MARK(X)
 A__TL(cons(X,Y)) -> MARK(Y)
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> A__HD(mark(X))
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> MARK(X)
 MARK(tl(X)) -> A__TL(mark(X))
 MARK(tl(X)) -> MARK(X)
->->-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros

Problem 1: 

Reduction Pairs Processor:
-> Pairs:
 A__HD(cons(X,Y)) -> MARK(X)
 A__TL(cons(X,Y)) -> MARK(Y)
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> A__HD(mark(X))
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> MARK(X)
 MARK(tl(X)) -> A__TL(mark(X))
 MARK(tl(X)) -> MARK(X)
-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros
-> Usable rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__adx](X) = X
[a__hd](X) = X + 2
[a__incr](X) = X
[a__nats] = 2
[a__tl](X) = 2.X + 2
[a__zeros] = 2
[mark](X) = 2.X + 2
[0] = 0
[adx](X) = X
[cons](X1,X2) = 2.X1 + 2.X2 + 2
[hd](X) = X + 2
[incr](X) = X
[nats] = 2
[s](X) = X
[tl](X) = 2.X + 2
[zeros] = 0
[A__HD](X) = X + 2
[A__TL](X) = 2.X + 2
[MARK](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A__TL(cons(X,Y)) -> MARK(Y)
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> A__HD(mark(X))
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> MARK(X)
 MARK(tl(X)) -> A__TL(mark(X))
 MARK(tl(X)) -> MARK(X)
-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__TL(cons(X,Y)) -> MARK(Y)
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> MARK(X)
 MARK(tl(X)) -> A__TL(mark(X))
 MARK(tl(X)) -> MARK(X)
->->-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros

Problem 1: 

Reduction Pairs Processor:
-> Pairs:
 A__TL(cons(X,Y)) -> MARK(Y)
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> MARK(X)
 MARK(tl(X)) -> A__TL(mark(X))
 MARK(tl(X)) -> MARK(X)
-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros
-> Usable rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__adx](X) = X
[a__hd](X) = X + 2
[a__incr](X) = X
[a__nats] = 2
[a__tl](X) = 2.X + 2
[a__zeros] = 2
[mark](X) = 2.X + 2
[0] = 0
[adx](X) = X
[cons](X1,X2) = 2.X1 + 2.X2 + 2
[hd](X) = X + 2
[incr](X) = X
[nats] = 0
[s](X) = X
[tl](X) = 2.X + 2
[zeros] = 0
[A__TL](X) = X + 2
[MARK](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> MARK(X)
 MARK(tl(X)) -> A__TL(mark(X))
 MARK(tl(X)) -> MARK(X)
-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> MARK(X)
 MARK(tl(X)) -> MARK(X)
->->-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros

Problem 1: 

Subterm Processor:
-> Pairs:
 MARK(adx(X)) -> MARK(X)
 MARK(hd(X)) -> MARK(X)
 MARK(incr(X)) -> MARK(X)
 MARK(tl(X)) -> MARK(X)
-> Rules:
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__adx(X) -> adx(X)
 a__hd(cons(X,Y)) -> mark(X)
 a__hd(X) -> hd(X)
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__incr(X) -> incr(X)
 a__nats -> a__adx(a__zeros)
 a__nats -> nats
 a__tl(cons(X,Y)) -> mark(Y)
 a__tl(X) -> tl(X)
 a__zeros -> cons(0,zeros)
 a__zeros -> zeros
 mark(0) -> 0
 mark(adx(X)) -> a__adx(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(hd(X)) -> a__hd(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(s(X)) -> s(X)
 mark(tl(X)) -> a__tl(mark(X))
 mark(zeros) -> a__zeros
->Projection:
 pi(MARK) = 1

Problem 1: 

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

The problem is finite.