YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
greaters(x:S,.(y:S,z:S)) -> if(<=(y:S,x:S),greaters(x:S,z:S),.(y:S,greaters(x:S,z:S)))
greaters(x:S,nil) -> nil
lowers(x:S,.(y:S,z:S)) -> if(<=(y:S,x:S),.(y:S,lowers(x:S,z:S)),lowers(x:S,z:S))
lowers(x:S,nil) -> nil
qsort(.(x:S,y:S)) -> ++(qsort(lowers(x:S,y:S)),.(x:S,qsort(greaters(x:S,y:S))))
qsort(nil) -> nil
)

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.3: 

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

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.