YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
f(x:S,y:S) -> g1(x:S,x:S,y:S)
f(x:S,y:S) -> g1(y:S,x:S,x:S)
f(x:S,y:S) -> g2(x:S,y:S,y:S)
f(x:S,y:S) -> g2(y:S,y:S,x:S)
g1(x:S,x:S,y:S) -> h(x:S,y:S)
g1(y:S,x:S,x:S) -> h(x:S,y:S)
g2(x:S,y:S,y:S) -> h(x:S,y:S)
g2(y:S,y:S,x:S) -> h(x:S,y:S)
h(x:S,x:S) -> x:S
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(x:S,y:S) -> G1(x:S,x:S,y:S)
 F(x:S,y:S) -> G1(y:S,x:S,x:S)
 F(x:S,y:S) -> G2(x:S,y:S,y:S)
 F(x:S,y:S) -> G2(y:S,y:S,x:S)
 G1(x:S,x:S,y:S) -> H(x:S,y:S)
 G1(y:S,x:S,x:S) -> H(x:S,y:S)
 G2(x:S,y:S,y:S) -> H(x:S,y:S)
 G2(y:S,y:S,x:S) -> H(x:S,y:S)
-> Rules:
 f(x:S,y:S) -> g1(x:S,x:S,y:S)
 f(x:S,y:S) -> g1(y:S,x:S,x:S)
 f(x:S,y:S) -> g2(x:S,y:S,y:S)
 f(x:S,y:S) -> g2(y:S,y:S,x:S)
 g1(x:S,x:S,y:S) -> h(x:S,y:S)
 g1(y:S,x:S,x:S) -> h(x:S,y:S)
 g2(x:S,y:S,y:S) -> h(x:S,y:S)
 g2(y:S,y:S,x:S) -> h(x:S,y:S)
 h(x:S,x:S) -> x:S

Problem 1: 

SCC Processor:
-> Pairs:
 F(x:S,y:S) -> G1(x:S,x:S,y:S)
 F(x:S,y:S) -> G1(y:S,x:S,x:S)
 F(x:S,y:S) -> G2(x:S,y:S,y:S)
 F(x:S,y:S) -> G2(y:S,y:S,x:S)
 G1(x:S,x:S,y:S) -> H(x:S,y:S)
 G1(y:S,x:S,x:S) -> H(x:S,y:S)
 G2(x:S,y:S,y:S) -> H(x:S,y:S)
 G2(y:S,y:S,x:S) -> H(x:S,y:S)
-> Rules:
 f(x:S,y:S) -> g1(x:S,x:S,y:S)
 f(x:S,y:S) -> g1(y:S,x:S,x:S)
 f(x:S,y:S) -> g2(x:S,y:S,y:S)
 f(x:S,y:S) -> g2(y:S,y:S,x:S)
 g1(x:S,x:S,y:S) -> h(x:S,y:S)
 g1(y:S,x:S,x:S) -> h(x:S,y:S)
 g2(x:S,y:S,y:S) -> h(x:S,y:S)
 g2(y:S,y:S,x:S) -> h(x:S,y:S)
 h(x:S,x:S) -> x:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.