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))
f(0) -> 0
f(s(0)) -> s(0)
f(s(s(x:S))) -> +(p(g(x:S)),q(g(x:S)))
f(s(s(x:S))) -> p(h(g(x:S)))
g(0) -> pair(s(0),s(0))
g(s(x:S)) -> h(g(x:S))
g(s(x:S)) -> pair(+(p(g(x:S)),q(g(x:S))),p(g(x:S)))
h(x:S) -> pair(+(p(x:S),q(x:S)),p(x:S))
p(pair(x:S,y:S)) -> x:S
q(pair(x:S,y:S)) -> y:S
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 F(s(s(x:S))) -> +#(p(g(x:S)),q(g(x:S)))
 F(s(s(x:S))) -> G(x:S)
 F(s(s(x:S))) -> H(g(x:S))
 F(s(s(x:S))) -> P(g(x:S))
 F(s(s(x:S))) -> P(h(g(x:S)))
 F(s(s(x:S))) -> Q(g(x:S))
 G(s(x:S)) -> +#(p(g(x:S)),q(g(x:S)))
 G(s(x:S)) -> G(x:S)
 G(s(x:S)) -> H(g(x:S))
 G(s(x:S)) -> P(g(x:S))
 G(s(x:S)) -> Q(g(x:S))
 H(x:S) -> +#(p(x:S),q(x:S))
 H(x:S) -> P(x:S)
 H(x:S) -> Q(x:S)
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x:S))) -> +(p(g(x:S)),q(g(x:S)))
 f(s(s(x:S))) -> p(h(g(x:S)))
 g(0) -> pair(s(0),s(0))
 g(s(x:S)) -> h(g(x:S))
 g(s(x:S)) -> pair(+(p(g(x:S)),q(g(x:S))),p(g(x:S)))
 h(x:S) -> pair(+(p(x:S),q(x:S)),p(x:S))
 p(pair(x:S,y:S)) -> x:S
 q(pair(x:S,y:S)) -> y:S

Problem 1: 

SCC Processor:
-> Pairs:
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 F(s(s(x:S))) -> +#(p(g(x:S)),q(g(x:S)))
 F(s(s(x:S))) -> G(x:S)
 F(s(s(x:S))) -> H(g(x:S))
 F(s(s(x:S))) -> P(g(x:S))
 F(s(s(x:S))) -> P(h(g(x:S)))
 F(s(s(x:S))) -> Q(g(x:S))
 G(s(x:S)) -> +#(p(g(x:S)),q(g(x:S)))
 G(s(x:S)) -> G(x:S)
 G(s(x:S)) -> H(g(x:S))
 G(s(x:S)) -> P(g(x:S))
 G(s(x:S)) -> Q(g(x:S))
 H(x:S) -> +#(p(x:S),q(x:S))
 H(x:S) -> P(x:S)
 H(x:S) -> Q(x:S)
-> Rules:
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x:S))) -> +(p(g(x:S)),q(g(x:S)))
 f(s(s(x:S))) -> p(h(g(x:S)))
 g(0) -> pair(s(0),s(0))
 g(s(x:S)) -> h(g(x:S))
 g(s(x:S)) -> pair(+(p(g(x:S)),q(g(x:S))),p(g(x:S)))
 h(x:S) -> pair(+(p(x:S),q(x:S)),p(x:S))
 p(pair(x:S,y:S)) -> x:S
 q(pair(x:S,y: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))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x:S))) -> +(p(g(x:S)),q(g(x:S)))
 f(s(s(x:S))) -> p(h(g(x:S)))
 g(0) -> pair(s(0),s(0))
 g(s(x:S)) -> h(g(x:S))
 g(s(x:S)) -> pair(+(p(g(x:S)),q(g(x:S))),p(g(x:S)))
 h(x:S) -> pair(+(p(x:S),q(x:S)),p(x:S))
 p(pair(x:S,y:S)) -> x:S
 q(pair(x:S,y: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))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x:S))) -> +(p(g(x:S)),q(g(x:S)))
 f(s(s(x:S))) -> p(h(g(x:S)))
 g(0) -> pair(s(0),s(0))
 g(s(x:S)) -> h(g(x:S))
 g(s(x:S)) -> pair(+(p(g(x:S)),q(g(x:S))),p(g(x:S)))
 h(x:S) -> pair(+(p(x:S),q(x:S)),p(x:S))
 p(pair(x:S,y:S)) -> x:S
 q(pair(x:S,y: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))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x:S))) -> +(p(g(x:S)),q(g(x:S)))
 f(s(s(x:S))) -> p(h(g(x:S)))
 g(0) -> pair(s(0),s(0))
 g(s(x:S)) -> h(g(x:S))
 g(s(x:S)) -> pair(+(p(g(x:S)),q(g(x:S))),p(g(x:S)))
 h(x:S) -> pair(+(p(x:S),q(x:S)),p(x:S))
 p(pair(x:S,y:S)) -> x:S
 q(pair(x:S,y: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))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x:S))) -> +(p(g(x:S)),q(g(x:S)))
 f(s(s(x:S))) -> p(h(g(x:S)))
 g(0) -> pair(s(0),s(0))
 g(s(x:S)) -> h(g(x:S))
 g(s(x:S)) -> pair(+(p(g(x:S)),q(g(x:S))),p(g(x:S)))
 h(x:S) -> pair(+(p(x:S),q(x:S)),p(x:S))
 p(pair(x:S,y:S)) -> x:S
 q(pair(x:S,y: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))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x:S))) -> +(p(g(x:S)),q(g(x:S)))
 f(s(s(x:S))) -> p(h(g(x:S)))
 g(0) -> pair(s(0),s(0))
 g(s(x:S)) -> h(g(x:S))
 g(s(x:S)) -> pair(+(p(g(x:S)),q(g(x:S))),p(g(x:S)))
 h(x:S) -> pair(+(p(x:S),q(x:S)),p(x:S))
 p(pair(x:S,y:S)) -> x:S
 q(pair(x:S,y: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))
 f(0) -> 0
 f(s(0)) -> s(0)
 f(s(s(x:S))) -> +(p(g(x:S)),q(g(x:S)))
 f(s(s(x:S))) -> p(h(g(x:S)))
 g(0) -> pair(s(0),s(0))
 g(s(x:S)) -> h(g(x:S))
 g(s(x:S)) -> pair(+(p(g(x:S)),q(g(x:S))),p(g(x:S)))
 h(x:S) -> pair(+(p(x:S),q(x:S)),p(x:S))
 p(pair(x:S,y:S)) -> x:S
 q(pair(x:S,y:S)) -> y:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.