YES

Problem 1: 

(VAR v_NonEmpty:S p:S x:S xs:S y:S)
(RULES
app(app(app(if,ttrue),x:S),y:S) -> x:S
app(app(app(if,ttrue),x:S),y:S) -> y:S
app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
app(app(dropWhile,p:S),nil) -> nil
app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
app(app(takeWhile,p:S),nil) -> nil
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(dropWhile,p:S),xs:S)
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S))
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(if,app(p:S,x:S))
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(p:S,x:S)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,x:S),app(app(takeWhile,p:S),xs:S))
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S)))
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(takeWhile,p:S),xs:S)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(if,app(p:S,x:S))
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(p:S,x:S)
-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(dropWhile,p:S),xs:S)
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S))
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(if,app(p:S,x:S))
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(p:S,x:S)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(cons,x:S),app(app(takeWhile,p:S),xs:S))
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S)))
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(takeWhile,p:S),xs:S)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(if,app(p:S,x:S))
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(p:S,x:S)
-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(dropWhile,p:S),xs:S)
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(p:S,x:S)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(takeWhile,p:S),xs:S)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(p:S,x:S)
->->-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil

Problem 1: 

Subterm Processor:
-> Pairs:
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(dropWhile,p:S),xs:S)
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(p:S,x:S)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(takeWhile,p:S),xs:S)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(p:S,x:S)
-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil
->Projection:
 pi(APP) = 1

Problem 1: 

SCC Processor:
-> Pairs:
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(dropWhile,p:S),xs:S)
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(takeWhile,p:S),xs:S)
-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(takeWhile,p:S),xs:S)
->->-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil
->->Cycle:
->->-> Pairs:
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(dropWhile,p:S),xs:S)
->->-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 APP(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(takeWhile,p:S),xs:S)
-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil
->Projection:
 pi(APP) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 APP(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> APP(app(dropWhile,p:S),xs:S)
-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil
->Projection:
 pi(APP) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 app(app(app(if,ttrue),x:S),y:S) -> x:S
 app(app(app(if,ttrue),x:S),y:S) -> y:S
 app(app(dropWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(dropWhile,p:S),xs:S)),app(app(cons,x:S),xs:S))
 app(app(dropWhile,p:S),nil) -> nil
 app(app(takeWhile,p:S),app(app(cons,x:S),xs:S)) -> app(app(app(if,app(p:S,x:S)),app(app(cons,x:S),app(app(takeWhile,p:S),xs:S))),nil)
 app(app(takeWhile,p:S),nil) -> nil
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.