YES

Problem 1: 

(VAR v_NonEmpty:S u:S x:S y:S z:S)
(RULES
f(j(x:S,y:S),y:S) -> g(f(x:S,k(y:S)))
f(x:S,h1(y:S,z:S)) -> h2(0,x:S,h1(y:S,z:S))
g(h2(x:S,y:S,h1(z:S,u:S))) -> h2(s(x:S),y:S,h1(z:S,u:S))
h2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> h2(s(x:S),y:S,h1(s(z:S),u:S))
i(f(x:S,h(y:S))) -> y:S
i(h2(s(x:S),y:S,h1(x:S,z:S))) -> z:S
k(h(x:S)) -> h1(0,x:S)
k(h1(x:S,y:S)) -> h1(s(x:S),y:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(j(x:S,y:S),y:S) -> F(x:S,k(y:S))
 F(j(x:S,y:S),y:S) -> G(f(x:S,k(y:S)))
 F(j(x:S,y:S),y:S) -> K(y:S)
 F(x:S,h1(y:S,z:S)) -> H2(0,x:S,h1(y:S,z:S))
 G(h2(x:S,y:S,h1(z:S,u:S))) -> H2(s(x:S),y:S,h1(z:S,u:S))
 H2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> H2(s(x:S),y:S,h1(s(z:S),u:S))
-> Rules:
 f(j(x:S,y:S),y:S) -> g(f(x:S,k(y:S)))
 f(x:S,h1(y:S,z:S)) -> h2(0,x:S,h1(y:S,z:S))
 g(h2(x:S,y:S,h1(z:S,u:S))) -> h2(s(x:S),y:S,h1(z:S,u:S))
 h2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> h2(s(x:S),y:S,h1(s(z:S),u:S))
 i(f(x:S,h(y:S))) -> y:S
 i(h2(s(x:S),y:S,h1(x:S,z:S))) -> z:S
 k(h(x:S)) -> h1(0,x:S)
 k(h1(x:S,y:S)) -> h1(s(x:S),y:S)

Problem 1: 

SCC Processor:
-> Pairs:
 F(j(x:S,y:S),y:S) -> F(x:S,k(y:S))
 F(j(x:S,y:S),y:S) -> G(f(x:S,k(y:S)))
 F(j(x:S,y:S),y:S) -> K(y:S)
 F(x:S,h1(y:S,z:S)) -> H2(0,x:S,h1(y:S,z:S))
 G(h2(x:S,y:S,h1(z:S,u:S))) -> H2(s(x:S),y:S,h1(z:S,u:S))
 H2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> H2(s(x:S),y:S,h1(s(z:S),u:S))
-> Rules:
 f(j(x:S,y:S),y:S) -> g(f(x:S,k(y:S)))
 f(x:S,h1(y:S,z:S)) -> h2(0,x:S,h1(y:S,z:S))
 g(h2(x:S,y:S,h1(z:S,u:S))) -> h2(s(x:S),y:S,h1(z:S,u:S))
 h2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> h2(s(x:S),y:S,h1(s(z:S),u:S))
 i(f(x:S,h(y:S))) -> y:S
 i(h2(s(x:S),y:S,h1(x:S,z:S))) -> z:S
 k(h(x:S)) -> h1(0,x:S)
 k(h1(x:S,y:S)) -> h1(s(x:S),y:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 H2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> H2(s(x:S),y:S,h1(s(z:S),u:S))
->->-> Rules:
 f(j(x:S,y:S),y:S) -> g(f(x:S,k(y:S)))
 f(x:S,h1(y:S,z:S)) -> h2(0,x:S,h1(y:S,z:S))
 g(h2(x:S,y:S,h1(z:S,u:S))) -> h2(s(x:S),y:S,h1(z:S,u:S))
 h2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> h2(s(x:S),y:S,h1(s(z:S),u:S))
 i(f(x:S,h(y:S))) -> y:S
 i(h2(s(x:S),y:S,h1(x:S,z:S))) -> z:S
 k(h(x:S)) -> h1(0,x:S)
 k(h1(x:S,y:S)) -> h1(s(x:S),y:S)
->->Cycle:
->->-> Pairs:
 F(j(x:S,y:S),y:S) -> F(x:S,k(y:S))
->->-> Rules:
 f(j(x:S,y:S),y:S) -> g(f(x:S,k(y:S)))
 f(x:S,h1(y:S,z:S)) -> h2(0,x:S,h1(y:S,z:S))
 g(h2(x:S,y:S,h1(z:S,u:S))) -> h2(s(x:S),y:S,h1(z:S,u:S))
 h2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> h2(s(x:S),y:S,h1(s(z:S),u:S))
 i(f(x:S,h(y:S))) -> y:S
 i(h2(s(x:S),y:S,h1(x:S,z:S))) -> z:S
 k(h(x:S)) -> h1(0,x:S)
 k(h1(x:S,y:S)) -> h1(s(x:S),y:S)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 H2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> H2(s(x:S),y:S,h1(s(z:S),u:S))
-> Rules:
 f(j(x:S,y:S),y:S) -> g(f(x:S,k(y:S)))
 f(x:S,h1(y:S,z:S)) -> h2(0,x:S,h1(y:S,z:S))
 g(h2(x:S,y:S,h1(z:S,u:S))) -> h2(s(x:S),y:S,h1(z:S,u:S))
 h2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> h2(s(x:S),y:S,h1(s(z:S),u:S))
 i(f(x:S,h(y:S))) -> y:S
 i(h2(s(x:S),y:S,h1(x:S,z:S))) -> z:S
 k(h(x:S)) -> h1(0,x:S)
 k(h1(x:S,y:S)) -> h1(s(x:S),y:S)
->Projection:
 pi(H2) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(j(x:S,y:S),y:S) -> g(f(x:S,k(y:S)))
 f(x:S,h1(y:S,z:S)) -> h2(0,x:S,h1(y:S,z:S))
 g(h2(x:S,y:S,h1(z:S,u:S))) -> h2(s(x:S),y:S,h1(z:S,u:S))
 h2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> h2(s(x:S),y:S,h1(s(z:S),u:S))
 i(f(x:S,h(y:S))) -> y:S
 i(h2(s(x:S),y:S,h1(x:S,z:S))) -> z:S
 k(h(x:S)) -> h1(0,x:S)
 k(h1(x:S,y:S)) -> h1(s(x:S),y:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 F(j(x:S,y:S),y:S) -> F(x:S,k(y:S))
-> Rules:
 f(j(x:S,y:S),y:S) -> g(f(x:S,k(y:S)))
 f(x:S,h1(y:S,z:S)) -> h2(0,x:S,h1(y:S,z:S))
 g(h2(x:S,y:S,h1(z:S,u:S))) -> h2(s(x:S),y:S,h1(z:S,u:S))
 h2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> h2(s(x:S),y:S,h1(s(z:S),u:S))
 i(f(x:S,h(y:S))) -> y:S
 i(h2(s(x:S),y:S,h1(x:S,z:S))) -> z:S
 k(h(x:S)) -> h1(0,x:S)
 k(h1(x:S,y:S)) -> h1(s(x:S),y:S)
->Projection:
 pi(F) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(j(x:S,y:S),y:S) -> g(f(x:S,k(y:S)))
 f(x:S,h1(y:S,z:S)) -> h2(0,x:S,h1(y:S,z:S))
 g(h2(x:S,y:S,h1(z:S,u:S))) -> h2(s(x:S),y:S,h1(z:S,u:S))
 h2(x:S,j(y:S,h1(z:S,u:S)),h1(z:S,u:S)) -> h2(s(x:S),y:S,h1(s(z:S),u:S))
 i(f(x:S,h(y:S))) -> y:S
 i(h2(s(x:S),y:S,h1(x:S,z:S))) -> z:S
 k(h(x:S)) -> h1(0,x:S)
 k(h1(x:S,y:S)) -> h1(s(x:S),y:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.