YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
+(0,y:S) -> y:S
+(s(x:S),y:S) -> s(+(x:S,y:S))
+(x:S,+(y:S,z:S)) -> +(+(x:S,y:S),z:S)
+(x:S,0) -> x:S
+(x:S,s(y:S)) -> s(+(x:S,y:S))
f(g(f(x:S))) -> f(h(s(0),x:S))
f(g(h(x:S,y:S))) -> f(h(s(x:S),y:S))
f(h(x:S,h(y:S,z:S))) -> f(h(+(x:S,y:S),z:S))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 +#(s(x:S),y:S) -> +#(x:S,y:S)
 +#(x:S,+(y:S,z:S)) -> +#(+(x:S,y:S),z:S)
 +#(x:S,+(y:S,z:S)) -> +#(x:S,y:S)
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 F(g(f(x:S))) -> F(h(s(0),x:S))
 F(g(h(x:S,y:S))) -> F(h(s(x:S),y:S))
 F(h(x:S,h(y:S,z:S))) -> +#(x:S,y:S)
 F(h(x:S,h(y:S,z:S))) -> F(h(+(x:S,y:S),z:S))
-> Rules:
 +(0,y:S) -> y:S
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,+(y:S,z:S)) -> +(+(x:S,y:S),z:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(g(f(x:S))) -> f(h(s(0),x:S))
 f(g(h(x:S,y:S))) -> f(h(s(x:S),y:S))
 f(h(x:S,h(y:S,z:S))) -> f(h(+(x:S,y:S),z:S))

Problem 1: 

SCC Processor:
-> Pairs:
 +#(s(x:S),y:S) -> +#(x:S,y:S)
 +#(x:S,+(y:S,z:S)) -> +#(+(x:S,y:S),z:S)
 +#(x:S,+(y:S,z:S)) -> +#(x:S,y:S)
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
 F(g(f(x:S))) -> F(h(s(0),x:S))
 F(g(h(x:S,y:S))) -> F(h(s(x:S),y:S))
 F(h(x:S,h(y:S,z:S))) -> +#(x:S,y:S)
 F(h(x:S,h(y:S,z:S))) -> F(h(+(x:S,y:S),z:S))
-> Rules:
 +(0,y:S) -> y:S
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,+(y:S,z:S)) -> +(+(x:S,y:S),z:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(g(f(x:S))) -> f(h(s(0),x:S))
 f(g(h(x:S,y:S))) -> f(h(s(x:S),y:S))
 f(h(x:S,h(y:S,z:S))) -> f(h(+(x:S,y:S),z:S))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 +#(s(x:S),y:S) -> +#(x:S,y:S)
 +#(x:S,+(y:S,z:S)) -> +#(+(x:S,y:S),z:S)
 +#(x:S,+(y:S,z:S)) -> +#(x:S,y:S)
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
->->-> Rules:
 +(0,y:S) -> y:S
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,+(y:S,z:S)) -> +(+(x:S,y:S),z:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(g(f(x:S))) -> f(h(s(0),x:S))
 f(g(h(x:S,y:S))) -> f(h(s(x:S),y:S))
 f(h(x:S,h(y:S,z:S))) -> f(h(+(x:S,y:S),z:S))
->->Cycle:
->->-> Pairs:
 F(h(x:S,h(y:S,z:S))) -> F(h(+(x:S,y:S),z:S))
->->-> Rules:
 +(0,y:S) -> y:S
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,+(y:S,z:S)) -> +(+(x:S,y:S),z:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(g(f(x:S))) -> f(h(s(0),x:S))
 f(g(h(x:S,y:S))) -> f(h(s(x:S),y:S))
 f(h(x:S,h(y:S,z:S))) -> f(h(+(x:S,y:S),z: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,+(y:S,z:S)) -> +#(+(x:S,y:S),z:S)
 +#(x:S,+(y:S,z:S)) -> +#(x:S,y:S)
 +#(x:S,s(y:S)) -> +#(x:S,y:S)
-> Rules:
 +(0,y:S) -> y:S
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,+(y:S,z:S)) -> +(+(x:S,y:S),z:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(g(f(x:S))) -> f(h(s(0),x:S))
 f(g(h(x:S,y:S))) -> f(h(s(x:S),y:S))
 f(h(x:S,h(y:S,z:S))) -> f(h(+(x:S,y:S),z:S))
->Projection:
 pi(+#) = 2

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 F(h(x:S,h(y:S,z:S))) -> F(h(+(x:S,y:S),z:S))
-> Rules:
 +(0,y:S) -> y:S
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,+(y:S,z:S)) -> +(+(x:S,y:S),z:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
 f(g(f(x:S))) -> f(h(s(0),x:S))
 f(g(h(x:S,y:S))) -> f(h(s(x:S),y:S))
 f(h(x:S,h(y:S,z:S))) -> f(h(+(x:S,y:S),z:S))
-> Usable rules:
 +(0,y:S) -> y:S
 +(s(x:S),y:S) -> s(+(x:S,y:S))
 +(x:S,+(y:S,z:S)) -> +(+(x:S,y:S),z:S)
 +(x:S,0) -> x:S
 +(x:S,s(y:S)) -> s(+(x:S,y:S))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[+](X1,X2) = X1 + 2.X2 + 1
[0] = 0
[h](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = 2
[F](X) = 2.X

Problem 1.2: 

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

The problem is finite.