YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
+(x:S,0) -> x:S
+(x:S,s(y:S)) -> s(+(x:S,y:S))
fib(0) -> 0
fib(s(0)) -> s(0)
fib(s(s(0))) -> s(0)
fib(s(s(x:S))) -> sp(g(x:S))
g(0) -> pair(s(0),0)
g(s(0)) -> pair(s(0),s(0))
g(s(x:S)) -> np(g(x:S))
np(pair(x:S,y:S)) -> pair(+(x:S,y:S),x:S)
sp(pair(x:S,y:S)) -> +(x:S,y:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 FIB(s(s(x:S))) -> G(x:S)
 FIB(s(s(x:S))) -> SP(g(x:S))
 G(s(x:S)) -> G(x:S)
 G(s(x:S)) -> NP(g(x:S))
 NP(pair(x:S,y:S)) -> +#(x:S,y:S)
 SP(pair(x:S,y:S)) -> +#(x:S,y:S)
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fib(0) -> 0
 fib(s(0)) -> s(0)
 fib(s(s(0))) -> s(0)
 fib(s(s(x:S))) -> sp(g(x:S))
 g(0) -> pair(s(0),0)
 g(s(0)) -> pair(s(0),s(0))
 g(s(x:S)) -> np(g(x:S))
 np(pair(x:S,y:S)) -> pair(+(x:S,y:S),x:S)
 sp(pair(x:S,y:S)) -> +(x:S,y:S)

Problem 1: 

SCC Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 FIB(s(s(x:S))) -> G(x:S)
 FIB(s(s(x:S))) -> SP(g(x:S))
 G(s(x:S)) -> G(x:S)
 G(s(x:S)) -> NP(g(x:S))
 NP(pair(x:S,y:S)) -> +#(x:S,y:S)
 SP(pair(x:S,y:S)) -> +#(x:S,y:S)
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fib(0) -> 0
 fib(s(0)) -> s(0)
 fib(s(s(0))) -> s(0)
 fib(s(s(x:S))) -> sp(g(x:S))
 g(0) -> pair(s(0),0)
 g(s(0)) -> pair(s(0),s(0))
 g(s(x:S)) -> np(g(x:S))
 np(pair(x:S,y:S)) -> pair(+(x:S,y:S),x:S)
 sp(pair(x:S,y:S)) -> +(x:S,y:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fib(0) -> 0
 fib(s(0)) -> s(0)
 fib(s(s(0))) -> s(0)
 fib(s(s(x:S))) -> sp(g(x:S))
 g(0) -> pair(s(0),0)
 g(s(0)) -> pair(s(0),s(0))
 g(s(x:S)) -> np(g(x:S))
 np(pair(x:S,y:S)) -> pair(+(x:S,y:S),x:S)
 sp(pair(x:S,y:S)) -> +(x:S,y:S)
->->Cycle:
->->-> Pairs:
 G(s(x:S)) -> G(x:S)
->->-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fib(0) -> 0
 fib(s(0)) -> s(0)
 fib(s(s(0))) -> s(0)
 fib(s(s(x:S))) -> sp(g(x:S))
 g(0) -> pair(s(0),0)
 g(s(0)) -> pair(s(0),s(0))
 g(s(x:S)) -> np(g(x:S))
 np(pair(x:S,y:S)) -> pair(+(x:S,y:S),x:S)
 sp(pair(x:S,y:S)) -> +(x:S,y:S)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fib(0) -> 0
 fib(s(0)) -> s(0)
 fib(s(s(0))) -> s(0)
 fib(s(s(x:S))) -> sp(g(x:S))
 g(0) -> pair(s(0),0)
 g(s(0)) -> pair(s(0),s(0))
 g(s(x:S)) -> np(g(x:S))
 np(pair(x:S,y:S)) -> pair(+(x:S,y:S),x:S)
 sp(pair(x:S,y:S)) -> +(x:S,y:S)
->Projection:
 pi(+#) = 2

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 G(s(x:S)) -> G(x:S)
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 fib(0) -> 0
 fib(s(0)) -> s(0)
 fib(s(s(0))) -> s(0)
 fib(s(s(x:S))) -> sp(g(x:S))
 g(0) -> pair(s(0),0)
 g(s(0)) -> pair(s(0),s(0))
 g(s(x:S)) -> np(g(x:S))
 np(pair(x:S,y:S)) -> pair(+(x:S,y:S),x:S)
 sp(pair(x:S,y:S)) -> +(x:S,y:S)
->Projection:
 pi(G) = 1

Problem 1.2: 

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

The problem is finite.