YES

Problem 1: 

(VAR v_NonEmpty:S L:S X:S X1:S X2:S Y:S)
(RULES
a__eq(0,0) -> ttrue
a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
a__eq(X:S,Y:S) -> ffalse
a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
a__inf(X:S) -> cons(X:S,inf(s(X:S)))
a__inf(X:S) -> inf(X:S)
a__length(cons(X:S,L:S)) -> s(length(L:S))
a__length(nil) -> 0
a__length(X:S) -> length(X:S)
a__take(0,X:S) -> nil
a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
a__take(X1:S,X2:S) -> take(X1:S,X2:S)
mark(0) -> 0
mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
mark(ffalse) -> ffalse
mark(inf(X:S)) -> a__inf(mark(X:S))
mark(length(X:S)) -> a__length(mark(X:S))
mark(nil) -> nil
mark(s(X:S)) -> s(X:S)
mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
mark(ttrue) -> ttrue
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A__EQ(s(X:S),s(Y:S)) -> A__EQ(X:S,Y:S)
 MARK(eq(X1:S,X2:S)) -> A__EQ(X1:S,X2:S)
 MARK(inf(X:S)) -> A__INF(mark(X:S))
 MARK(inf(X:S)) -> MARK(X:S)
 MARK(length(X:S)) -> A__LENGTH(mark(X:S))
 MARK(length(X:S)) -> MARK(X:S)
 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S))
 MARK(take(X1:S,X2:S)) -> MARK(X1:S)
 MARK(take(X1:S,X2:S)) -> MARK(X2:S)
-> Rules:
 a__eq(0,0) -> ttrue
 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
 a__eq(X:S,Y:S) -> ffalse
 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
 a__inf(X:S) -> inf(X:S)
 a__length(cons(X:S,L:S)) -> s(length(L:S))
 a__length(nil) -> 0
 a__length(X:S) -> length(X:S)
 a__take(0,X:S) -> nil
 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
 mark(ffalse) -> ffalse
 mark(inf(X:S)) -> a__inf(mark(X:S))
 mark(length(X:S)) -> a__length(mark(X:S))
 mark(nil) -> nil
 mark(s(X:S)) -> s(X:S)
 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
 mark(ttrue) -> ttrue

Problem 1: 

SCC Processor:
-> Pairs:
 A__EQ(s(X:S),s(Y:S)) -> A__EQ(X:S,Y:S)
 MARK(eq(X1:S,X2:S)) -> A__EQ(X1:S,X2:S)
 MARK(inf(X:S)) -> A__INF(mark(X:S))
 MARK(inf(X:S)) -> MARK(X:S)
 MARK(length(X:S)) -> A__LENGTH(mark(X:S))
 MARK(length(X:S)) -> MARK(X:S)
 MARK(take(X1:S,X2:S)) -> A__TAKE(mark(X1:S),mark(X2:S))
 MARK(take(X1:S,X2:S)) -> MARK(X1:S)
 MARK(take(X1:S,X2:S)) -> MARK(X2:S)
-> Rules:
 a__eq(0,0) -> ttrue
 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
 a__eq(X:S,Y:S) -> ffalse
 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
 a__inf(X:S) -> inf(X:S)
 a__length(cons(X:S,L:S)) -> s(length(L:S))
 a__length(nil) -> 0
 a__length(X:S) -> length(X:S)
 a__take(0,X:S) -> nil
 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
 mark(ffalse) -> ffalse
 mark(inf(X:S)) -> a__inf(mark(X:S))
 mark(length(X:S)) -> a__length(mark(X:S))
 mark(nil) -> nil
 mark(s(X:S)) -> s(X:S)
 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
 mark(ttrue) -> ttrue
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A__EQ(s(X:S),s(Y:S)) -> A__EQ(X:S,Y:S)
->->-> Rules:
 a__eq(0,0) -> ttrue
 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
 a__eq(X:S,Y:S) -> ffalse
 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
 a__inf(X:S) -> inf(X:S)
 a__length(cons(X:S,L:S)) -> s(length(L:S))
 a__length(nil) -> 0
 a__length(X:S) -> length(X:S)
 a__take(0,X:S) -> nil
 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
 mark(ffalse) -> ffalse
 mark(inf(X:S)) -> a__inf(mark(X:S))
 mark(length(X:S)) -> a__length(mark(X:S))
 mark(nil) -> nil
 mark(s(X:S)) -> s(X:S)
 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
 mark(ttrue) -> ttrue
->->Cycle:
->->-> Pairs:
 MARK(inf(X:S)) -> MARK(X:S)
 MARK(length(X:S)) -> MARK(X:S)
 MARK(take(X1:S,X2:S)) -> MARK(X1:S)
 MARK(take(X1:S,X2:S)) -> MARK(X2:S)
->->-> Rules:
 a__eq(0,0) -> ttrue
 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
 a__eq(X:S,Y:S) -> ffalse
 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
 a__inf(X:S) -> inf(X:S)
 a__length(cons(X:S,L:S)) -> s(length(L:S))
 a__length(nil) -> 0
 a__length(X:S) -> length(X:S)
 a__take(0,X:S) -> nil
 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
 mark(ffalse) -> ffalse
 mark(inf(X:S)) -> a__inf(mark(X:S))
 mark(length(X:S)) -> a__length(mark(X:S))
 mark(nil) -> nil
 mark(s(X:S)) -> s(X:S)
 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
 mark(ttrue) -> ttrue


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 A__EQ(s(X:S),s(Y:S)) -> A__EQ(X:S,Y:S)
-> Rules:
 a__eq(0,0) -> ttrue
 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
 a__eq(X:S,Y:S) -> ffalse
 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
 a__inf(X:S) -> inf(X:S)
 a__length(cons(X:S,L:S)) -> s(length(L:S))
 a__length(nil) -> 0
 a__length(X:S) -> length(X:S)
 a__take(0,X:S) -> nil
 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
 mark(ffalse) -> ffalse
 mark(inf(X:S)) -> a__inf(mark(X:S))
 mark(length(X:S)) -> a__length(mark(X:S))
 mark(nil) -> nil
 mark(s(X:S)) -> s(X:S)
 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
 mark(ttrue) -> ttrue
->Projection:
 pi(A__EQ) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a__eq(0,0) -> ttrue
 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
 a__eq(X:S,Y:S) -> ffalse
 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
 a__inf(X:S) -> inf(X:S)
 a__length(cons(X:S,L:S)) -> s(length(L:S))
 a__length(nil) -> 0
 a__length(X:S) -> length(X:S)
 a__take(0,X:S) -> nil
 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
 mark(ffalse) -> ffalse
 mark(inf(X:S)) -> a__inf(mark(X:S))
 mark(length(X:S)) -> a__length(mark(X:S))
 mark(nil) -> nil
 mark(s(X:S)) -> s(X:S)
 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
 mark(ttrue) -> ttrue
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 MARK(inf(X:S)) -> MARK(X:S)
 MARK(length(X:S)) -> MARK(X:S)
 MARK(take(X1:S,X2:S)) -> MARK(X1:S)
 MARK(take(X1:S,X2:S)) -> MARK(X2:S)
-> Rules:
 a__eq(0,0) -> ttrue
 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
 a__eq(X:S,Y:S) -> ffalse
 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
 a__inf(X:S) -> inf(X:S)
 a__length(cons(X:S,L:S)) -> s(length(L:S))
 a__length(nil) -> 0
 a__length(X:S) -> length(X:S)
 a__take(0,X:S) -> nil
 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
 mark(ffalse) -> ffalse
 mark(inf(X:S)) -> a__inf(mark(X:S))
 mark(length(X:S)) -> a__length(mark(X:S))
 mark(nil) -> nil
 mark(s(X:S)) -> s(X:S)
 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
 mark(ttrue) -> ttrue
->Projection:
 pi(MARK) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a__eq(0,0) -> ttrue
 a__eq(s(X:S),s(Y:S)) -> a__eq(X:S,Y:S)
 a__eq(X:S,Y:S) -> ffalse
 a__eq(X1:S,X2:S) -> eq(X1:S,X2:S)
 a__inf(X:S) -> cons(X:S,inf(s(X:S)))
 a__inf(X:S) -> inf(X:S)
 a__length(cons(X:S,L:S)) -> s(length(L:S))
 a__length(nil) -> 0
 a__length(X:S) -> length(X:S)
 a__take(0,X:S) -> nil
 a__take(s(X:S),cons(Y:S,L:S)) -> cons(Y:S,take(X:S,L:S))
 a__take(X1:S,X2:S) -> take(X1:S,X2:S)
 mark(0) -> 0
 mark(cons(X1:S,X2:S)) -> cons(X1:S,X2:S)
 mark(eq(X1:S,X2:S)) -> a__eq(X1:S,X2:S)
 mark(ffalse) -> ffalse
 mark(inf(X:S)) -> a__inf(mark(X:S))
 mark(length(X:S)) -> a__length(mark(X:S))
 mark(nil) -> nil
 mark(s(X:S)) -> s(X:S)
 mark(take(X1:S,X2:S)) -> a__take(mark(X1:S),mark(X2:S))
 mark(ttrue) -> ttrue
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.