YES

Problem 1: 

(VAR v_NonEmpty:S L:S M:S N:S X:S Y:S)
(RULES
app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
app(nil,Y:S) -> Y:S
high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
high(N:S,nil) -> nil
ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
le(0,Y:S) -> ttrue
le(s(X:S),0) -> ffalse
le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
low(N:S,nil) -> nil
quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
quicksort(nil) -> nil
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(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:
 APP(cons(N:S,L:S),Y:S) -> APP(L:S,Y:S)
 HIGH(N:S,cons(M:S,L:S)) -> IFHIGH(le(M:S,N:S),N:S,cons(M:S,L:S))
 HIGH(N:S,cons(M:S,L:S)) -> LE(M:S,N:S)
 IFHIGH(ffalse,N:S,cons(M:S,L:S)) -> HIGH(N:S,L:S)
 IFHIGH(ttrue,N:S,cons(M:S,L:S)) -> HIGH(N:S,L:S)
 IFLOW(ffalse,N:S,cons(M:S,L:S)) -> LOW(N:S,L:S)
 IFLOW(ttrue,N:S,cons(M:S,L:S)) -> LOW(N:S,L:S)
 LE(s(X:S),s(Y:S)) -> LE(X:S,Y:S)
 LOW(N:S,cons(M:S,L:S)) -> IFLOW(le(M:S,N:S),N:S,cons(M:S,L:S))
 LOW(N:S,cons(M:S,L:S)) -> LE(M:S,N:S)
 QUICKSORT(cons(N:S,L:S)) -> APP(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 QUICKSORT(cons(N:S,L:S)) -> HIGH(N:S,L:S)
 QUICKSORT(cons(N:S,L:S)) -> LOW(N:S,L:S)
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(high(N:S,L:S))
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(low(N:S,L:S))
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 APP(cons(N:S,L:S),Y:S) -> APP(L:S,Y:S)
 HIGH(N:S,cons(M:S,L:S)) -> IFHIGH(le(M:S,N:S),N:S,cons(M:S,L:S))
 HIGH(N:S,cons(M:S,L:S)) -> LE(M:S,N:S)
 IFHIGH(ffalse,N:S,cons(M:S,L:S)) -> HIGH(N:S,L:S)
 IFHIGH(ttrue,N:S,cons(M:S,L:S)) -> HIGH(N:S,L:S)
 IFLOW(ffalse,N:S,cons(M:S,L:S)) -> LOW(N:S,L:S)
 IFLOW(ttrue,N:S,cons(M:S,L:S)) -> LOW(N:S,L:S)
 LE(s(X:S),s(Y:S)) -> LE(X:S,Y:S)
 LOW(N:S,cons(M:S,L:S)) -> IFLOW(le(M:S,N:S),N:S,cons(M:S,L:S))
 LOW(N:S,cons(M:S,L:S)) -> LE(M:S,N:S)
 QUICKSORT(cons(N:S,L:S)) -> APP(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 QUICKSORT(cons(N:S,L:S)) -> HIGH(N:S,L:S)
 QUICKSORT(cons(N:S,L:S)) -> LOW(N:S,L:S)
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(high(N:S,L:S))
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(low(N:S,L:S))
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 LE(s(X:S),s(Y:S)) -> LE(X:S,Y:S)
->->-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->->Cycle:
->->-> Pairs:
 IFLOW(ffalse,N:S,cons(M:S,L:S)) -> LOW(N:S,L:S)
 IFLOW(ttrue,N:S,cons(M:S,L:S)) -> LOW(N:S,L:S)
 LOW(N:S,cons(M:S,L:S)) -> IFLOW(le(M:S,N:S),N:S,cons(M:S,L:S))
->->-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->->Cycle:
->->-> Pairs:
 HIGH(N:S,cons(M:S,L:S)) -> IFHIGH(le(M:S,N:S),N:S,cons(M:S,L:S))
 IFHIGH(ffalse,N:S,cons(M:S,L:S)) -> HIGH(N:S,L:S)
 IFHIGH(ttrue,N:S,cons(M:S,L:S)) -> HIGH(N:S,L:S)
->->-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->->Cycle:
->->-> Pairs:
 APP(cons(N:S,L:S),Y:S) -> APP(L:S,Y:S)
->->-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->->Cycle:
->->-> Pairs:
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(high(N:S,L:S))
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(low(N:S,L:S))
->->-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil


The problem is decomposed in 5 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 LE(s(X:S),s(Y:S)) -> LE(X:S,Y:S)
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Projection:
 pi(LE) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 IFLOW(ffalse,N:S,cons(M:S,L:S)) -> LOW(N:S,L:S)
 IFLOW(ttrue,N:S,cons(M:S,L:S)) -> LOW(N:S,L:S)
 LOW(N:S,cons(M:S,L:S)) -> IFLOW(le(M:S,N:S),N:S,cons(M:S,L:S))
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Projection:
 pi(IFLOW) = 3
 pi(LOW) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 LOW(N:S,cons(M:S,L:S)) -> IFLOW(le(M:S,N:S),N:S,cons(M:S,L:S))
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 HIGH(N:S,cons(M:S,L:S)) -> IFHIGH(le(M:S,N:S),N:S,cons(M:S,L:S))
 IFHIGH(ffalse,N:S,cons(M:S,L:S)) -> HIGH(N:S,L:S)
 IFHIGH(ttrue,N:S,cons(M:S,L:S)) -> HIGH(N:S,L:S)
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Projection:
 pi(HIGH) = 2
 pi(IFHIGH) = 3

Problem 1.3: 

SCC Processor:
-> Pairs:
 HIGH(N:S,cons(M:S,L:S)) -> IFHIGH(le(M:S,N:S),N:S,cons(M:S,L:S))
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 APP(cons(N:S,L:S),Y:S) -> APP(L:S,Y:S)
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Projection:
 pi(APP) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Reduction Pairs Processor:
-> Pairs:
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(high(N:S,L:S))
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(low(N:S,L:S))
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
-> Usable rules:
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 0
[high](X1,X2) = 2.X2 + 1
[ifhigh](X1,X2,X3) = 2.X3
[iflow](X1,X2,X3) = 2.X3 + 1
[le](X1,X2) = X1 + 2.X2 + 2
[low](X1,X2) = 2.X2 + 1
[quicksort](X) = 0
[0] = 2
[cons](X1,X2) = 2.X1 + 2.X2 + 2
[fSNonEmpty] = 0
[false] = 2
[nil] = 2
[s](X) = 2.X + 2
[true] = 2
[APP](X1,X2) = 0
[HIGH](X1,X2) = 0
[IFHIGH](X1,X2,X3) = 0
[IFLOW](X1,X2,X3) = 0
[LE](X1,X2) = 0
[LOW](X1,X2) = 0
[QUICKSORT](X) = 2.X

Problem 1.5: 

SCC Processor:
-> Pairs:
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(low(N:S,L:S))
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(low(N:S,L:S))
->->-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil

Problem 1.5: 

Reduction Pairs Processor:
-> Pairs:
 QUICKSORT(cons(N:S,L:S)) -> QUICKSORT(low(N:S,L:S))
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
-> Usable rules:
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 0
[high](X1,X2) = 0
[ifhigh](X1,X2,X3) = 0
[iflow](X1,X2,X3) = 2.X3 + 1
[le](X1,X2) = 2.X1 + 2.X2 + 2
[low](X1,X2) = 2.X2 + 1
[quicksort](X) = 0
[0] = 1
[cons](X1,X2) = 2.X1 + 2.X2 + 2
[fSNonEmpty] = 0
[false] = 0
[nil] = 2
[s](X) = 2.X
[true] = 0
[APP](X1,X2) = 0
[HIGH](X1,X2) = 0
[IFHIGH](X1,X2,X3) = 0
[IFLOW](X1,X2,X3) = 0
[LE](X1,X2) = 0
[LOW](X1,X2) = 0
[QUICKSORT](X) = 2.X

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(cons(N:S,L:S),Y:S) -> cons(N:S,app(L:S,Y:S))
 app(nil,Y:S) -> Y:S
 high(N:S,cons(M:S,L:S)) -> ifhigh(le(M:S,N:S),N:S,cons(M:S,L:S))
 high(N:S,nil) -> nil
 ifhigh(ffalse,N:S,cons(M:S,L:S)) -> cons(M:S,high(N:S,L:S))
 ifhigh(ttrue,N:S,cons(M:S,L:S)) -> high(N:S,L:S)
 iflow(ffalse,N:S,cons(M:S,L:S)) -> low(N:S,L:S)
 iflow(ttrue,N:S,cons(M:S,L:S)) -> cons(M:S,low(N:S,L:S))
 le(0,Y:S) -> ttrue
 le(s(X:S),0) -> ffalse
 le(s(X:S),s(Y:S)) -> le(X:S,Y:S)
 low(N:S,cons(M:S,L:S)) -> iflow(le(M:S,N:S),N:S,cons(M:S,L:S))
 low(N:S,nil) -> nil
 quicksort(cons(N:S,L:S)) -> app(quicksort(low(N:S,L:S)),cons(N:S,quicksort(high(N:S,L:S))))
 quicksort(nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.