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


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YES

Problem 1: 

(VAR X X1 X2 XS)
(RULES
a__head(cons(X,XS)) -> mark(X)
a__head(X) -> head(X)
a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
a__incr(X) -> incr(X)
a__nats -> cons(0,incr(nats))
a__nats -> nats
a__odds -> a__incr(a__pairs)
a__odds -> odds
a__pairs -> cons(0,incr(odds))
a__pairs -> pairs
a__tail(cons(X,XS)) -> mark(XS)
a__tail(X) -> tail(X)
mark(0) -> 0
mark(cons(X1,X2)) -> cons(mark(X1),X2)
mark(head(X)) -> a__head(mark(X))
mark(incr(X)) -> a__incr(mark(X))
mark(nats) -> a__nats
mark(nil) -> nil
mark(odds) -> a__odds
mark(pairs) -> a__pairs
mark(s(X)) -> s(mark(X))
mark(tail(X)) -> a__tail(mark(X))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__HEAD(cons(X,XS)) -> MARK(X)
 A__INCR(cons(X,XS)) -> MARK(X)
 A__ODDS -> A__INCR(a__pairs)
 A__ODDS -> A__PAIRS
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> A__HEAD(mark(X))
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(nats) -> A__NATS
 MARK(odds) -> A__ODDS
 MARK(pairs) -> A__PAIRS
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))

Problem 1: 

SCC Processor:
-> Pairs:
 A__HEAD(cons(X,XS)) -> MARK(X)
 A__INCR(cons(X,XS)) -> MARK(X)
 A__ODDS -> A__INCR(a__pairs)
 A__ODDS -> A__PAIRS
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> A__HEAD(mark(X))
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(nats) -> A__NATS
 MARK(odds) -> A__ODDS
 MARK(pairs) -> A__PAIRS
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__HEAD(cons(X,XS)) -> MARK(X)
 A__INCR(cons(X,XS)) -> MARK(X)
 A__ODDS -> A__INCR(a__pairs)
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> A__HEAD(mark(X))
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(odds) -> A__ODDS
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
->->-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A__HEAD(cons(X,XS)) -> MARK(X)
 A__INCR(cons(X,XS)) -> MARK(X)
 A__ODDS -> A__INCR(a__pairs)
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> A__HEAD(mark(X))
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(odds) -> A__ODDS
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
-> Usable rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__head](X) = 2.X + 2
[a__incr](X) = 2.X
[a__nats] = 0
[a__odds] = 0
[a__pairs] = 0
[a__tail](X) = 2.X + 2
[mark](X) = 2.X
[0] = 0
[cons](X1,X2) = 2.X1 + 2.X2
[head](X) = 2.X + 2
[incr](X) = 2.X
[nats] = 0
[nil] = 2
[odds] = 0
[pairs] = 0
[s](X) = X
[tail](X) = 2.X + 2
[A__HEAD](X) = 2.X + 2
[A__INCR](X) = X + 1
[A__ODDS] = 1
[A__TAIL](X) = X + 2
[MARK](X) = 2.X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 A__ODDS -> A__INCR(a__pairs)
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> A__HEAD(mark(X))
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(odds) -> A__ODDS
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 A__ODDS -> A__INCR(a__pairs)
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(odds) -> A__ODDS
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
->->-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 A__ODDS -> A__INCR(a__pairs)
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(odds) -> A__ODDS
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
-> Usable rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__head](X) = 2.X + 2
[a__incr](X) = X
[a__nats] = 2
[a__odds] = 1
[a__pairs] = 1
[a__tail](X) = 2.X + 2
[mark](X) = X
[0] = 0
[cons](X1,X2) = 2.X1 + X2
[head](X) = 2.X + 2
[incr](X) = X
[nats] = 2
[nil] = 1
[odds] = 1
[pairs] = 1
[s](X) = X
[tail](X) = 2.X + 2
[A__INCR](X) = X
[A__ODDS] = 2
[A__TAIL](X) = 2.X + 2
[MARK](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(odds) -> A__ODDS
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
->->-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 A__TAIL(cons(X,XS)) -> MARK(XS)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
-> Usable rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__head](X) = X + 2
[a__incr](X) = 2.X
[a__nats] = 2
[a__odds] = 0
[a__pairs] = 0
[a__tail](X) = 2.X + 2
[mark](X) = 2.X
[0] = 0
[cons](X1,X2) = 2.X1 + X2
[head](X) = X + 2
[incr](X) = 2.X
[nats] = 1
[nil] = 1
[odds] = 0
[pairs] = 0
[s](X) = X
[tail](X) = 2.X + 2
[A__INCR](X) = X + 1
[A__TAIL](X) = X + 2
[MARK](X) = X + 1

Problem 1: 

SCC Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> A__TAIL(mark(X))
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> MARK(X)
->->-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(head(X)) -> MARK(X)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
-> Usable rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__head](X) = 2.X + 2
[a__incr](X) = 2.X
[a__nats] = 0
[a__odds] = 0
[a__pairs] = 0
[a__tail](X) = 2.X
[mark](X) = 2.X
[0] = 0
[cons](X1,X2) = 2.X1 + 2.X2
[head](X) = 2.X + 2
[incr](X) = 2.X
[nats] = 0
[nil] = 2
[odds] = 0
[pairs] = 0
[s](X) = X
[tail](X) = 2.X
[A__INCR](X) = 2.X + 2
[MARK](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> MARK(X)
->->-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
 MARK(tail(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
-> Usable rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__head](X) = 2.X + 2
[a__incr](X) = 2.X
[a__nats] = 2
[a__odds] = 0
[a__pairs] = 0
[a__tail](X) = 2.X + 2
[mark](X) = 2.X
[0] = 0
[cons](X1,X2) = 2.X1 + X2
[head](X) = 2.X + 2
[incr](X) = 2.X
[nats] = 1
[nil] = 2
[odds] = 0
[pairs] = 0
[s](X) = X
[tail](X) = 2.X + 1
[A__INCR](X) = 2.X + 2
[MARK](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
->->-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A__INCR(cons(X,XS)) -> MARK(X)
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
-> Usable rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a__head](X) = 2.X + 1
[a__incr](X) = X
[a__nats] = 2
[a__odds] = 2
[a__pairs] = 2
[a__tail](X) = 2.X + 2
[mark](X) = X
[0] = 0
[cons](X1,X2) = 2.X1 + 1/2.X2 + 1
[head](X) = 2.X + 1
[incr](X) = X
[nats] = 2
[nil] = 2
[odds] = 2
[pairs] = 2
[s](X) = X
[tail](X) = 2.X + 2
[A__INCR](X) = 2.X + 2
[MARK](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(incr(X)) -> A__INCR(mark(X))
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
->->-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))

Problem 1: 

Subterm Processor:
-> Pairs:
 MARK(cons(X1,X2)) -> MARK(X1)
 MARK(incr(X)) -> MARK(X)
 MARK(s(X)) -> MARK(X)
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Projection:
 pi(MARK) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a__head(cons(X,XS)) -> mark(X)
 a__head(X) -> head(X)
 a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS))
 a__incr(X) -> incr(X)
 a__nats -> cons(0,incr(nats))
 a__nats -> nats
 a__odds -> a__incr(a__pairs)
 a__odds -> odds
 a__pairs -> cons(0,incr(odds))
 a__pairs -> pairs
 a__tail(cons(X,XS)) -> mark(XS)
 a__tail(X) -> tail(X)
 mark(0) -> 0
 mark(cons(X1,X2)) -> cons(mark(X1),X2)
 mark(head(X)) -> a__head(mark(X))
 mark(incr(X)) -> a__incr(mark(X))
 mark(nats) -> a__nats
 mark(nil) -> nil
 mark(odds) -> a__odds
 mark(pairs) -> a__pairs
 mark(s(X)) -> s(mark(X))
 mark(tail(X)) -> a__tail(mark(X))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.