YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
f(0) -> ttrue
f(1) -> ffalse
f(s(x:S)) -> f(x:S)
g(x:S,c(y:S)) -> g(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
g(x:S,c(y:S)) -> c(g(x:S,y:S))
if(ffalse,s(x:S),s(y:S)) -> s(y:S)
if(ttrue,s(x:S),s(y:S)) -> s(x:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(s(x:S)) -> F(x:S)
 G(x:S,c(y:S)) -> F(x:S)
 G(x:S,c(y:S)) -> G(s(x:S),y:S)
 G(x:S,c(y:S)) -> G(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 G(x:S,c(y:S)) -> G(x:S,y:S)
 G(x:S,c(y:S)) -> IF(f(x:S),c(g(s(x:S),y:S)),c(y:S))
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(x:S,c(y:S)) -> g(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 g(x:S,c(y:S)) -> c(g(x:S,y:S))
 if(ffalse,s(x:S),s(y:S)) -> s(y:S)
 if(ttrue,s(x:S),s(y:S)) -> s(x:S)

Problem 1: 

SCC Processor:
-> Pairs:
 F(s(x:S)) -> F(x:S)
 G(x:S,c(y:S)) -> F(x:S)
 G(x:S,c(y:S)) -> G(s(x:S),y:S)
 G(x:S,c(y:S)) -> G(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 G(x:S,c(y:S)) -> G(x:S,y:S)
 G(x:S,c(y:S)) -> IF(f(x:S),c(g(s(x:S),y:S)),c(y:S))
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(x:S,c(y:S)) -> g(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 g(x:S,c(y:S)) -> c(g(x:S,y:S))
 if(ffalse,s(x:S),s(y:S)) -> s(y:S)
 if(ttrue,s(x:S),s(y:S)) -> s(x:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(s(x:S)) -> F(x:S)
->->-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(x:S,c(y:S)) -> g(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 g(x:S,c(y:S)) -> c(g(x:S,y:S))
 if(ffalse,s(x:S),s(y:S)) -> s(y:S)
 if(ttrue,s(x:S),s(y:S)) -> s(x:S)
->->Cycle:
->->-> Pairs:
 G(x:S,c(y:S)) -> G(s(x:S),y:S)
 G(x:S,c(y:S)) -> G(x:S,y:S)
->->-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(x:S,c(y:S)) -> g(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 g(x:S,c(y:S)) -> c(g(x:S,y:S))
 if(ffalse,s(x:S),s(y:S)) -> s(y:S)
 if(ttrue,s(x:S),s(y:S)) -> s(x:S)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 F(s(x:S)) -> F(x:S)
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(x:S,c(y:S)) -> g(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 g(x:S,c(y:S)) -> c(g(x:S,y:S))
 if(ffalse,s(x:S),s(y:S)) -> s(y:S)
 if(ttrue,s(x:S),s(y:S)) -> s(x:S)
->Projection:
 pi(F) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(x:S,c(y:S)) -> g(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 g(x:S,c(y:S)) -> c(g(x:S,y:S))
 if(ffalse,s(x:S),s(y:S)) -> s(y:S)
 if(ttrue,s(x:S),s(y:S)) -> s(x:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 G(x:S,c(y:S)) -> G(s(x:S),y:S)
 G(x:S,c(y:S)) -> G(x:S,y:S)
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(x:S,c(y:S)) -> g(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 g(x:S,c(y:S)) -> c(g(x:S,y:S))
 if(ffalse,s(x:S),s(y:S)) -> s(y:S)
 if(ttrue,s(x:S),s(y:S)) -> s(x:S)
->Projection:
 pi(G) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 f(0) -> ttrue
 f(1) -> ffalse
 f(s(x:S)) -> f(x:S)
 g(x:S,c(y:S)) -> g(x:S,if(f(x:S),c(g(s(x:S),y:S)),c(y:S)))
 g(x:S,c(y:S)) -> c(g(x:S,y:S))
 if(ffalse,s(x:S),s(y:S)) -> s(y:S)
 if(ttrue,s(x:S),s(y:S)) -> s(x:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.