YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
-(s(x:S),s(y:S)) -> -(x:S,y:S)
-(x:S,0) -> x:S
double(0) -> 0
double(s(x:S)) -> s(s(double(x:S)))
half(double(x:S)) -> x:S
half(0) -> 0
half(s(0)) -> 0
half(s(s(x:S))) -> s(half(x:S))
if(0,y:S,z:S) -> y:S
if(s(x:S),y:S,z:S) -> z:S
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
 DOUBLE(s(x:S)) -> DOUBLE(x:S)
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 half(double(x:S)) -> x:S
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(0,y:S,z:S) -> y:S
 if(s(x:S),y:S,z:S) -> z:S

Problem 1: 

SCC Processor:
-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
 DOUBLE(s(x:S)) -> DOUBLE(x:S)
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 half(double(x:S)) -> x:S
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(0,y:S,z:S) -> y:S
 if(s(x:S),y:S,z:S) -> z:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
->->-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 half(double(x:S)) -> x:S
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(0,y:S,z:S) -> y:S
 if(s(x:S),y:S,z:S) -> z:S
->->Cycle:
->->-> Pairs:
 DOUBLE(s(x:S)) -> DOUBLE(x:S)
->->-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 half(double(x:S)) -> x:S
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(0,y:S,z:S) -> y:S
 if(s(x:S),y:S,z:S) -> z:S
->->Cycle:
->->-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
->->-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 half(double(x:S)) -> x:S
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(0,y:S,z:S) -> y:S
 if(s(x:S),y:S,z:S) -> z:S


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 HALF(s(s(x:S))) -> HALF(x:S)
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 half(double(x:S)) -> x:S
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(0,y:S,z:S) -> y:S
 if(s(x:S),y:S,z:S) -> z:S
->Projection:
 pi(HALF) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 DOUBLE(s(x:S)) -> DOUBLE(x:S)
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 half(double(x:S)) -> x:S
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(0,y:S,z:S) -> y:S
 if(s(x:S),y:S,z:S) -> z:S
->Projection:
 pi(DOUBLE) = 1

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 -#(s(x:S),s(y:S)) -> -#(x:S,y:S)
-> Rules:
 -(s(x:S),s(y:S)) -> -(x:S,y:S)
 -(x:S,0) -> x:S
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 half(double(x:S)) -> x:S
 half(0) -> 0
 half(s(0)) -> 0
 half(s(s(x:S))) -> s(half(x:S))
 if(0,y:S,z:S) -> y:S
 if(s(x:S),y:S,z:S) -> z:S
->Projection:
 pi(-#) = 1

Problem 1.3: 

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

The problem is finite.