YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
a(y:S,0) -> b(y:S,0)
b(b(0,y:S),x:S) -> y:S
c(c(c(y:S))) -> c(c(a(a(c(b(0,y:S)),0),0)))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(y:S,0) -> B(y:S,0)
 C(c(c(y:S))) -> A(a(c(b(0,y:S)),0),0)
 C(c(c(y:S))) -> A(c(b(0,y:S)),0)
 C(c(c(y:S))) -> C(a(a(c(b(0,y:S)),0),0))
 C(c(c(y:S))) -> C(c(a(a(c(b(0,y:S)),0),0)))
-> Rules:
 a(y:S,0) -> b(y:S,0)
 b(b(0,y:S),x:S) -> y:S
 c(c(c(y:S))) -> c(c(a(a(c(b(0,y:S)),0),0)))

Problem 1: 

SCC Processor:
-> Pairs:
 A(y:S,0) -> B(y:S,0)
 C(c(c(y:S))) -> A(a(c(b(0,y:S)),0),0)
 C(c(c(y:S))) -> A(c(b(0,y:S)),0)
 C(c(c(y:S))) -> C(a(a(c(b(0,y:S)),0),0))
 C(c(c(y:S))) -> C(c(a(a(c(b(0,y:S)),0),0)))
-> Rules:
 a(y:S,0) -> b(y:S,0)
 b(b(0,y:S),x:S) -> y:S
 c(c(c(y:S))) -> c(c(a(a(c(b(0,y:S)),0),0)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 C(c(c(y:S))) -> C(a(a(c(b(0,y:S)),0),0))
 C(c(c(y:S))) -> C(c(a(a(c(b(0,y:S)),0),0)))
->->-> Rules:
 a(y:S,0) -> b(y:S,0)
 b(b(0,y:S),x:S) -> y:S
 c(c(c(y:S))) -> c(c(a(a(c(b(0,y:S)),0),0)))

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 C(c(c(y:S))) -> C(a(a(c(b(0,y:S)),0),0))
 C(c(c(y:S))) -> C(c(a(a(c(b(0,y:S)),0),0)))
-> Rules:
 a(y:S,0) -> b(y:S,0)
 b(b(0,y:S),x:S) -> y:S
 c(c(c(y:S))) -> c(c(a(a(c(b(0,y:S)),0),0)))
-> Usable rules:
 a(y:S,0) -> b(y:S,0)
 b(b(0,y:S),x:S) -> y:S
 c(c(c(y:S))) -> c(c(a(a(c(b(0,y:S)),0),0)))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a](X1,X2) = X1
[b](X1,X2) = X1 + X2
[c](X) = 2.X + 2
[0] = 0
[C](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 C(c(c(y:S))) -> C(c(a(a(c(b(0,y:S)),0),0)))
-> Rules:
 a(y:S,0) -> b(y:S,0)
 b(b(0,y:S),x:S) -> y:S
 c(c(c(y:S))) -> c(c(a(a(c(b(0,y:S)),0),0)))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 C(c(c(y:S))) -> C(c(a(a(c(b(0,y:S)),0),0)))
->->-> Rules:
 a(y:S,0) -> b(y:S,0)
 b(b(0,y:S),x:S) -> y:S
 c(c(c(y:S))) -> c(c(a(a(c(b(0,y:S)),0),0)))

Problem 1: 

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

Problem 1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 a(y:S,0) -> b(y:S,0)
 b(b(0,y:S),x:S) -> y:S
 c(c(c(y:S))) -> c(c(a(a(c(b(0,y:S)),0),0)))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.