YES

Problem 1: 

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

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(f(x:S)) -> F(g(f(x:S),x:S))
 F(f(x:S)) -> F(h(f(x:S),f(x:S)))
 F(f(x:S)) -> G(f(x:S),x:S)
 F(f(x:S)) -> H(f(x:S),f(x:S))
 H(x:S,x:S) -> G(x:S,0)
-> Rules:
 f(f(x:S)) -> f(g(f(x:S),x:S))
 f(f(x:S)) -> f(h(f(x:S),f(x:S)))
 g(x:S,y:S) -> y:S
 h(x:S,x:S) -> g(x:S,0)

Problem 1: 

SCC Processor:
-> Pairs:
 F(f(x:S)) -> F(g(f(x:S),x:S))
 F(f(x:S)) -> F(h(f(x:S),f(x:S)))
 F(f(x:S)) -> G(f(x:S),x:S)
 F(f(x:S)) -> H(f(x:S),f(x:S))
 H(x:S,x:S) -> G(x:S,0)
-> Rules:
 f(f(x:S)) -> f(g(f(x:S),x:S))
 f(f(x:S)) -> f(h(f(x:S),f(x:S)))
 g(x:S,y:S) -> y:S
 h(x:S,x:S) -> g(x:S,0)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(f(x:S)) -> F(g(f(x:S),x:S))
 F(f(x:S)) -> F(h(f(x:S),f(x:S)))
->->-> Rules:
 f(f(x:S)) -> f(g(f(x:S),x:S))
 f(f(x:S)) -> f(h(f(x:S),f(x:S)))
 g(x:S,y:S) -> y:S
 h(x:S,x:S) -> g(x:S,0)

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.