YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
+(s(x:S),y:S) -> s(+(x:S,y:S))
+(x:S,0) -> x:S
+(x:S,s(y:S)) -> s(+(x:S,y:S))
double(0) -> 0
double(s(x:S)) -> s(s(double(x:S)))
double(x:S) -> +(x:S,x:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(s(x:S),y:S) -> +#(x:S,y:S)
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 DOUBLE(s(x:S)) -> DOUBLE(x:S)
 DOUBLE(x:S) -> +#(x:S,x:S)
-> Rules:
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 double(x:S) -> +(x:S,x:S)

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

SCC Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 double(x:S) -> +(x:S,x:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 double(x:S) -> +(x:S,x:S)

Problem 1.1: 

Subterm Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 double(x:S) -> +(x:S,x:S)
->Projection:
 pi(+#) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 double(x:S) -> +(x:S,x: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),y:S) -> s(+(x:S,y:S))
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 double(0) -> 0
 double(s(x:S)) -> s(s(double(x:S)))
 double(x:S) -> +(x:S,x:S)
->Projection:
 pi(DOUBLE) = 1

Problem 1.2: 

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

The problem is finite.