YES

Problem 1: 

(VAR v_NonEmpty:S x:S)
(RULES
bits(0) -> 0
bits(s(x:S)) -> s(bits(half(s(x:S))))
half(0) -> 0
half(s(0)) -> 0
half(s(s(x:S))) -> s(half(x:S))
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 BITS(s(x:S)) -> BITS(half(s(x:S)))
 BITS(s(x:S)) -> HALF(s(x:S))
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 bits(0) -> 0
 bits(s(x:S)) -> s(bits(half(s(x:S))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))

Problem 1: 

SCC Processor:
-> Pairs:
 BITS(s(x:S)) -> BITS(half(s(x:S)))
 BITS(s(x:S)) -> HALF(s(x:S))
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 bits(0) -> 0
 bits(s(x:S)) -> s(bits(half(s(x:S))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
->->-> Rules:
 bits(0) -> 0
 bits(s(x:S)) -> s(bits(half(s(x:S))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->->Cycle:
->->-> Pairs:
 BITS(s(x:S)) -> BITS(half(s(x:S)))
->->-> Rules:
 bits(0) -> 0
 bits(s(x:S)) -> s(bits(half(s(x:S))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 bits(0) -> 0
 bits(s(x:S)) -> s(bits(half(s(x:S))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->Projection:
 pi(HALF) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 bits(0) -> 0
 bits(s(x:S)) -> s(bits(half(s(x:S))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 BITS(s(x:S)) -> BITS(half(s(x:S)))
-> Rules:
 bits(0) -> 0
 bits(s(x:S)) -> s(bits(half(s(x:S))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
-> Usable rules:
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[bits](X) = 0
[half](X) = 1/2.X
[0] = 0
[fSNonEmpty] = 0
[s](X) = X + 1/2
[BITS](X) = X
[HALF](X) = 0

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 bits(0) -> 0
 bits(s(x:S)) -> s(bits(half(s(x:S))))
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.