YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
c(b(a,a),b(y:S,z:S),x:S) -> b(a,b(z:S,z:S))
f(c(c(a,y:S,a),b(x:S,z:S),a)) -> b(y:S,f(c(f(a),z:S,z:S)))
f(b(b(x:S,f(y:S)),z:S)) -> c(z:S,x:S,f(b(b(f(a),y:S),y:S)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 F(b(b(x:S,f(y:S)),z:S)) -> C(z:S,x:S,f(b(b(f(a),y:S),y:S)))
 F(b(b(x:S,f(y:S)),z:S)) -> F(b(b(f(a),y:S),y:S))
-> Rules:
 c(b(a,a),b(y:S,z:S),x:S) -> b(a,b(z:S,z:S))
 f(c(c(a,y:S,a),b(x:S,z:S),a)) -> b(y:S,f(c(f(a),z:S,z:S)))
 f(b(b(x:S,f(y:S)),z:S)) -> c(z:S,x:S,f(b(b(f(a),y:S),y:S)))

Problem 1: 

SCC Processor:
-> Pairs:
 F(b(b(x:S,f(y:S)),z:S)) -> C(z:S,x:S,f(b(b(f(a),y:S),y:S)))
 F(b(b(x:S,f(y:S)),z:S)) -> F(b(b(f(a),y:S),y:S))
-> Rules:
 c(b(a,a),b(y:S,z:S),x:S) -> b(a,b(z:S,z:S))
 f(c(c(a,y:S,a),b(x:S,z:S),a)) -> b(y:S,f(c(f(a),z:S,z:S)))
 f(b(b(x:S,f(y:S)),z:S)) -> c(z:S,x:S,f(b(b(f(a),y:S),y:S)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 F(b(b(x:S,f(y:S)),z:S)) -> F(b(b(f(a),y:S),y:S))
->->-> Rules:
 c(b(a,a),b(y:S,z:S),x:S) -> b(a,b(z:S,z:S))
 f(c(c(a,y:S,a),b(x:S,z:S),a)) -> b(y:S,f(c(f(a),z:S,z:S)))
 f(b(b(x:S,f(y:S)),z:S)) -> c(z:S,x:S,f(b(b(f(a),y:S),y:S)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 F(b(b(x:S,f(y:S)),z:S)) -> F(b(b(f(a),y:S),y:S))
-> Rules:
 c(b(a,a),b(y:S,z:S),x:S) -> b(a,b(z:S,z:S))
 f(c(c(a,y:S,a),b(x:S,z:S),a)) -> b(y:S,f(c(f(a),z:S,z:S)))
 f(b(b(x:S,f(y:S)),z:S)) -> c(z:S,x:S,f(b(b(f(a),y:S),y:S)))
-> Usable rules:
 c(b(a,a),b(y:S,z:S),x:S) -> b(a,b(z:S,z:S))
 f(c(c(a,y:S,a),b(x:S,z:S),a)) -> b(y:S,f(c(f(a),z:S,z:S)))
 f(b(b(x:S,f(y:S)),z:S)) -> c(z:S,x:S,f(b(b(f(a),y:S),y:S)))
->Interpretation type:
 Linear
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 4
->Interpretation:
 
[c](X1,X2,X3) = 3.X1 + 4.X2 + 1/2
[f](X) = 4.X + 1/4
[a] = 0
[b](X1,X2) = 4/3.X1 + 2.X2 + 3/4
[F](X) = 3/2.X

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 c(b(a,a),b(y:S,z:S),x:S) -> b(a,b(z:S,z:S))
 f(c(c(a,y:S,a),b(x:S,z:S),a)) -> b(y:S,f(c(f(a),z:S,z:S)))
 f(b(b(x:S,f(y:S)),z:S)) -> c(z:S,x:S,f(b(b(f(a),y:S),y:S)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.