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


--------------------------------------------------------------------------------


YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 4 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.2: 

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

The problem is finite.

Problem 1.4: 

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

Problem 1.4: 

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

The problem is finite.