NO

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Narrowing Processor:
-> Pairs:
 H(f(f(x:S))) -> H(f(g(f(x:S))))
-> Rules:
 f(g(f(x:S))) -> f(f(x:S))
 h(f(f(x:S))) -> h(f(g(f(x:S))))
->Narrowed Pairs:
->->Original Pair:
 H(f(f(x:S))) -> H(f(g(f(x:S))))
->-> Narrowed pairs:
 H(f(f(g(f(x:S))))) -> H(f(g(f(f(x:S)))))
 H(f(f(x:S))) -> H(f(f(x:S)))

Problem 1: 

Infinite Processor:
-> Pairs:
 H(f(f(g(f(x:S))))) -> H(f(g(f(f(x:S)))))
 H(f(f(x:S))) -> H(f(f(x:S)))
-> Rules:
 f(g(f(x:S))) -> f(f(x:S))
 h(f(f(x:S))) -> h(f(g(f(x:S))))
-> Pairs in cycle:
 H(f(f(x:S))) -> H(f(f(x:S)))

The problem is infinite.