YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

The problem is finite.