YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
A(a(x1:S)) -> x1:S
A(c(b(x1:S))) -> b(c(A(x1:S)))
B(C(a(x1:S))) -> a(C(B(x1:S)))
B(b(x1:S)) -> x1:S
C(B(A(x1:S))) -> A(B(C(x1:S)))
C(c(x1:S)) -> x1:S
a(A(x1:S)) -> x1:S
a(b(c(x1:S))) -> c(b(a(x1:S)))
b(B(x1:S)) -> x1:S
b(a(C(x1:S))) -> C(a(b(x1:S)))
c(A(B(x1:S))) -> B(A(c(x1:S)))
c(C(x1:S)) -> x1:S
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A#(c(b(x1:S))) -> A#(x1:S)
 A#(c(b(x1:S))) -> b#(c(A(x1:S)))
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S

Problem 1: 

SCC Processor:
-> Pairs:
 A#(c(b(x1:S))) -> A#(x1:S)
 A#(c(b(x1:S))) -> b#(c(A(x1:S)))
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A#(c(b(x1:S))) -> A#(x1:S)
 A#(c(b(x1:S))) -> b#(c(A(x1:S)))
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A#(c(b(x1:S))) -> A#(x1:S)
 A#(c(b(x1:S))) -> b#(c(A(x1:S)))
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
-> Usable rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 1
[C](X) = 2.X + 2
[a](X) = 2.X + 1
[b](X) = 2.X + 2
[c](X) = 2.X + 1
[A#](X) = 2.X + 2
[B#](X) = 2.X + 2
[C#](X) = 2.X + 2
[a#](X) = 2.X + 2
[b#](X) = 2.X + 2
[c#](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 A#(c(b(x1:S))) -> b#(c(A(x1:S)))
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A#(c(b(x1:S))) -> b#(c(A(x1:S)))
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A#(c(b(x1:S))) -> b#(c(A(x1:S)))
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
-> Usable rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = X
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = X
[b](X) = 2.X + 1
[c](X) = 2.X + 1
[A#](X) = X
[B#](X) = 2.X + 2
[C#](X) = 2.X + 2
[a#](X) = X
[b#](X) = 2.X
[c#](X) = 2.X

Problem 1: 

SCC Processor:
-> Pairs:
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A#(c(b(x1:S))) -> c#(A(x1:S))
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
-> Usable rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[A#](X) = X + 2
[B#](X) = 2.X + 2
[C#](X) = 2.X + 2
[a#](X) = 2.X + 2
[b#](X) = 2.X + 2
[c#](X) = 2.X + 2

Problem 1: 

SCC Processor:
-> Pairs:
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> A#(B(C(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> A#(c(x1:S))
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 B#(C(a(x1:S))) -> B#(x1:S)
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
-> Usable rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = X
[c](X) = X
[B#](X) = 2.X + 2
[C#](X) = 2.X + 2
[a#](X) = 2.X + 2
[b#](X) = X
[c#](X) = X

Problem 1: 

SCC Processor:
-> Pairs:
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 B#(C(a(x1:S))) -> C#(B(x1:S))
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
-> Usable rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = X
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = X
[B#](X) = X
[C#](X) = 2.X
[a#](X) = 2.X + 2
[b#](X) = 2.X
[c#](X) = X

Problem 1: 

SCC Processor:
-> Pairs:
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 B#(C(a(x1:S))) -> a#(C(B(x1:S)))
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
-> Usable rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = X
[a](X) = X + 2
[b](X) = X + 2
[c](X) = X
[B#](X) = 2.X + 2
[C#](X) = X
[a#](X) = X + 2
[b#](X) = X + 2
[c#](X) = X

Problem 1: 

SCC Processor:
-> Pairs:
 C#(B(A(x1:S))) -> B#(C(x1:S))
 C#(B(A(x1:S))) -> C#(x1:S)
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 a#(b(c(x1:S))) -> c#(b(a(x1:S)))
 b#(a(C(x1:S))) -> C#(a(b(x1:S)))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
 c#(A(B(x1:S))) -> B#(A(c(x1:S)))
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 c#(A(B(x1:S))) -> c#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->->Cycle:
->->-> Pairs:
 C#(B(A(x1:S))) -> C#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->->Cycle:
->->-> Pairs:
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 c#(A(B(x1:S))) -> c#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Projection:
 pi(c#) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 C#(B(A(x1:S))) -> C#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Projection:
 pi(C#) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 a#(b(c(x1:S))) -> a#(x1:S)
 a#(b(c(x1:S))) -> b#(a(x1:S))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
-> Usable rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = X + 2
[B](X) = X + 2
[C](X) = X + 1
[a](X) = X + 2
[b](X) = X + 2
[c](X) = X
[a#](X) = 2.X + 2
[b#](X) = 2.X + 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 a#(b(c(x1:S))) -> b#(a(x1:S))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 a#(b(c(x1:S))) -> b#(a(x1:S))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S

Problem 1.3: 

Reduction Pair Processor:
-> Pairs:
 a#(b(c(x1:S))) -> b#(a(x1:S))
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
-> Usable rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[A](X) = 2.X + 2
[B](X) = 2.X + 2
[C](X) = 2.X + 2
[a](X) = 2.X + 2
[b](X) = 2.X + 2
[c](X) = 2.X + 2
[a#](X) = 2.X + 1
[b#](X) = 2.X + 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 b#(a(C(x1:S))) -> a#(b(x1:S))
 b#(a(C(x1:S))) -> b#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 b#(a(C(x1:S))) -> b#(x1:S)
->->-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S

Problem 1.3: 

Subterm Processor:
-> Pairs:
 b#(a(C(x1:S))) -> b#(x1:S)
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Projection:
 pi(b#) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 A(a(x1:S)) -> x1:S
 A(c(b(x1:S))) -> b(c(A(x1:S)))
 B(C(a(x1:S))) -> a(C(B(x1:S)))
 B(b(x1:S)) -> x1:S
 C(B(A(x1:S))) -> A(B(C(x1:S)))
 C(c(x1:S)) -> x1:S
 a(A(x1:S)) -> x1:S
 a(b(c(x1:S))) -> c(b(a(x1:S)))
 b(B(x1:S)) -> x1:S
 b(a(C(x1:S))) -> C(a(b(x1:S)))
 c(A(B(x1:S))) -> B(A(c(x1:S)))
 c(C(x1:S)) -> x1:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.