YES

Problem 1: 

(VAR v_NonEmpty:S x1:S)
(RULES
a(b(a(a(x1:S)))) -> b(a(b(a(x1:S))))
a(s(x1:S)) -> s(a(x1:S))
b(a(b(b(x1:S)))) -> a(b(a(b(x1:S))))
b(a(b(s(x1:S)))) -> a(b(s(a(x1:S))))
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 A(b(a(a(x1:S)))) -> A(b(a(x1:S)))
 A(b(a(a(x1:S)))) -> B(a(b(a(x1:S))))
 A(b(a(a(x1:S)))) -> B(a(x1:S))
 A(s(x1:S)) -> A(x1:S)
 B(a(b(b(x1:S)))) -> A(b(a(b(x1:S))))
 B(a(b(b(x1:S)))) -> A(b(x1:S))
 B(a(b(b(x1:S)))) -> B(a(b(x1:S)))
 B(a(b(s(x1:S)))) -> A(b(s(a(x1:S))))
 B(a(b(s(x1:S)))) -> A(x1:S)
 B(a(b(s(x1:S)))) -> B(s(a(x1:S)))
-> Rules:
 a(b(a(a(x1:S)))) -> b(a(b(a(x1:S))))
 a(s(x1:S)) -> s(a(x1:S))
 b(a(b(b(x1:S)))) -> a(b(a(b(x1:S))))
 b(a(b(s(x1:S)))) -> a(b(s(a(x1:S))))

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 A(b(a(a(x1:S)))) -> B(a(b(a(x1:S))))
 B(a(b(b(x1:S)))) -> A(b(a(b(x1:S))))
-> Rules:
 a(b(a(a(x1:S)))) -> b(a(b(a(x1:S))))
 a(s(x1:S)) -> s(a(x1:S))
 b(a(b(b(x1:S)))) -> a(b(a(b(x1:S))))
 b(a(b(s(x1:S)))) -> a(b(s(a(x1:S))))
-> Usable rules:
 a(b(a(a(x1:S)))) -> b(a(b(a(x1:S))))
 a(s(x1:S)) -> s(a(x1:S))
 b(a(b(b(x1:S)))) -> a(b(a(b(x1:S))))
 b(a(b(s(x1:S)))) -> a(b(s(a(x1:S))))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 2
->Bound:
 1
->Interpretation:
 
[a](X) = [1 0;1 0].X + [1;0]
[b](X) = [0 1;0 1].X + [1;1]
[s](X) = [1 0;0 0].X + [1;1]
[A](X) = [1 1;1 1].X + [0;1]
[B](X) = [1 1;1 1].X + [0;1]

Problem 1: 

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

The problem is finite.