/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 k l x y)
(RULES
app(cons(x,l),k) -> cons(x,app(l,k))
app(nil,k) -> k
app(l,nil) -> l
plus(0,y) -> y
plus(s(x),y) -> s(plus(x,y))
sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
sum(cons(x,nil)) -> cons(x,nil)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(cons(x,l),k) -> APP(l,k)
 PLUS(s(x),y) -> PLUS(x,y)
 SUM(app(l,cons(x,cons(y,k)))) -> APP(l,sum(cons(x,cons(y,k))))
 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k)))))
 SUM(app(l,cons(x,cons(y,k)))) -> SUM(cons(x,cons(y,k)))
 SUM(cons(x,cons(y,l))) -> PLUS(x,y)
 SUM(cons(x,cons(y,l))) -> SUM(cons(plus(x,y),l))
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)

Problem 1: 

SCC Processor:
-> Pairs:
 APP(cons(x,l),k) -> APP(l,k)
 PLUS(s(x),y) -> PLUS(x,y)
 SUM(app(l,cons(x,cons(y,k)))) -> APP(l,sum(cons(x,cons(y,k))))
 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k)))))
 SUM(app(l,cons(x,cons(y,k)))) -> SUM(cons(x,cons(y,k)))
 SUM(cons(x,cons(y,l))) -> PLUS(x,y)
 SUM(cons(x,cons(y,l))) -> SUM(cons(plus(x,y),l))
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(s(x),y) -> PLUS(x,y)
->->-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->->Cycle:
->->-> Pairs:
 SUM(cons(x,cons(y,l))) -> SUM(cons(plus(x,y),l))
->->-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->->Cycle:
->->-> Pairs:
 APP(cons(x,l),k) -> APP(l,k)
->->-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->->Cycle:
->->-> Pairs:
 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k)))))
->->-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 PLUS(s(x),y) -> PLUS(x,y)
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->Projection:
 pi(PLUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 SUM(cons(x,cons(y,l))) -> SUM(cons(plus(x,y),l))
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
-> Usable rules:
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[plus](X1,X2) = X2
[0] = 0
[cons](X1,X2) = 2.X1 + X2 + 2
[s](X) = X
[SUM](X) = 2.X

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 APP(cons(x,l),k) -> APP(l,k)
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->Projection:
 pi(APP) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pair Processor:
-> Pairs:
 SUM(app(l,cons(x,cons(y,k)))) -> SUM(app(l,sum(cons(x,cons(y,k)))))
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
-> Usable rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 2.X1 + 2.X2 + 1
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[sum](X) = 2
[0] = 0
[cons](X1,X2) = X2 + 2
[nil] = 0
[s](X) = X + 2
[SUM](X) = 2.X

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(cons(x,l),k) -> cons(x,app(l,k))
 app(nil,k) -> k
 app(l,nil) -> l
 plus(0,y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(cons(x,nil)) -> cons(x,nil)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.