YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 2 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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


The problem is decomposed in 2 subproblems.

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

Problem 1.2.1: 

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

The problem is finite.

Problem 1.2.2: 

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

Problem 1.2.2: 

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

The problem is finite.