/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 N X X1 X2 XS Y YS)
(RULES
activate(n__from(X)) -> from(X)
activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
activate(X) -> X
from(X) -> cons(X,n__from(s(X)))
from(X) -> n__from(X)
minus(s(X),s(Y)) -> minus(X,Y)
minus(X,0) -> 0
quot(0,s(Y)) -> 0
quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
sel(0,cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
zWquot(nil,XS) -> nil
zWquot(X1,X2) -> n__zWquot(X1,X2)
zWquot(XS,nil) -> nil
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 ACTIVATE(n__from(X)) -> FROM(X)
 ACTIVATE(n__zWquot(X1,X2)) -> ZWQUOT(X1,X2)
 MINUS(s(X),s(Y)) -> MINUS(X,Y)
 QUOT(s(X),s(Y)) -> MINUS(X,Y)
 QUOT(s(X),s(Y)) -> QUOT(minus(X,Y),s(Y))
 SEL(s(N),cons(X,XS)) -> ACTIVATE(XS)
 SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(XS)
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(YS)
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> QUOT(X,Y)
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__from(X)) -> FROM(X)
 ACTIVATE(n__zWquot(X1,X2)) -> ZWQUOT(X1,X2)
 MINUS(s(X),s(Y)) -> MINUS(X,Y)
 QUOT(s(X),s(Y)) -> MINUS(X,Y)
 QUOT(s(X),s(Y)) -> QUOT(minus(X,Y),s(Y))
 SEL(s(N),cons(X,XS)) -> ACTIVATE(XS)
 SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(XS)
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(YS)
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> QUOT(X,Y)
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(X),s(Y)) -> MINUS(X,Y)
->->-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->->Cycle:
->->-> Pairs:
 QUOT(s(X),s(Y)) -> QUOT(minus(X,Y),s(Y))
->->-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__zWquot(X1,X2)) -> ZWQUOT(X1,X2)
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(XS)
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(YS)
->->-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->->Cycle:
->->-> Pairs:
 SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
->->-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS(s(X),s(Y)) -> MINUS(X,Y)
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->Projection:
 pi(MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 QUOT(s(X),s(Y)) -> QUOT(minus(X,Y),s(Y))
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
-> Usable rules:
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[minus](X1,X2) = 2.X1 + 1
[0] = 0
[s](X) = 2.X + 2
[QUOT](X1,X2) = 2.X1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__zWquot(X1,X2)) -> ZWQUOT(X1,X2)
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(XS)
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(YS)
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons](X1,X2) = 2.X2
[n__zWquot](X1,X2) = X1 + X2 + 2
[ACTIVATE](X) = 2.X + 2
[ZWQUOT](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(XS)
 ZWQUOT(cons(X,XS),cons(Y,YS)) -> ACTIVATE(YS)
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 SEL(s(N),cons(X,XS)) -> SEL(N,activate(XS))
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->Projection:
 pi(SEL) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 activate(n__from(X)) -> from(X)
 activate(n__zWquot(X1,X2)) -> zWquot(X1,X2)
 activate(X) -> X
 from(X) -> cons(X,n__from(s(X)))
 from(X) -> n__from(X)
 minus(s(X),s(Y)) -> minus(X,Y)
 minus(X,0) -> 0
 quot(0,s(Y)) -> 0
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
 sel(0,cons(X,XS)) -> X
 sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
 zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
 zWquot(nil,XS) -> nil
 zWquot(X1,X2) -> n__zWquot(X1,X2)
 zWquot(XS,nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.