/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 y z)
(RULES
greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
greaters(x,nil) -> nil
lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
lowers(x,nil) -> nil
qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
qsort(nil) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 GREATERS(x,.(y,z)) -> GREATERS(x,z)
 LOWERS(x,.(y,z)) -> LOWERS(x,z)
 QSORT(.(x,y)) -> GREATERS(x,y)
 QSORT(.(x,y)) -> LOWERS(x,y)
 QSORT(.(x,y)) -> QSORT(greaters(x,y))
 QSORT(.(x,y)) -> QSORT(lowers(x,y))
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 GREATERS(x,.(y,z)) -> GREATERS(x,z)
 LOWERS(x,.(y,z)) -> LOWERS(x,z)
 QSORT(.(x,y)) -> GREATERS(x,y)
 QSORT(.(x,y)) -> LOWERS(x,y)
 QSORT(.(x,y)) -> QSORT(greaters(x,y))
 QSORT(.(x,y)) -> QSORT(lowers(x,y))
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 LOWERS(x,.(y,z)) -> LOWERS(x,z)
->->-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
->->Cycle:
->->-> Pairs:
 GREATERS(x,.(y,z)) -> GREATERS(x,z)
->->-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
->->Cycle:
->->-> Pairs:
 QSORT(.(x,y)) -> QSORT(greaters(x,y))
 QSORT(.(x,y)) -> QSORT(lowers(x,y))
->->-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 LOWERS(x,.(y,z)) -> LOWERS(x,z)
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
->Projection:
 pi(LOWERS) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 GREATERS(x,.(y,z)) -> GREATERS(x,z)
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
->Projection:
 pi(GREATERS) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 QSORT(.(x,y)) -> QSORT(greaters(x,y))
 QSORT(.(x,y)) -> QSORT(lowers(x,y))
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
-> Usable rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[greaters](X1,X2) = 2.X2 + 1
[lowers](X1,X2) = 2.X1 + 2
[.](X1,X2) = 2.X1 + 2.X2 + 2
[<=](X1,X2) = 0
[if](X1,X2,X3) = 2
[nil] = 2
[QSORT](X) = X

Problem 1.3: 

SCC Processor:
-> Pairs:
 QSORT(.(x,y)) -> QSORT(lowers(x,y))
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 QSORT(.(x,y)) -> QSORT(lowers(x,y))
->->-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 QSORT(.(x,y)) -> QSORT(lowers(x,y))
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
-> Usable rules:
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[lowers](X1,X2) = X1 + 2.X2
[.](X1,X2) = 2.X1 + 2.X2 + 1
[<=](X1,X2) = 2.X1
[if](X1,X2,X3) = 2.X1 + X3 + 1
[nil] = 2
[QSORT](X) = 2.X

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 greaters(x,.(y,z)) -> if(<=(y,x),greaters(x,z),.(y,greaters(x,z)))
 greaters(x,nil) -> nil
 lowers(x,.(y,z)) -> if(<=(y,x),.(y,lowers(x,z)),lowers(x,z))
 lowers(x,nil) -> nil
 qsort(.(x,y)) -> ++(qsort(lowers(x,y)),.(x,qsort(greaters(x,y))))
 qsort(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.