YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

Subterm Processor:
-> Pairs:
 B(c(x1:S)) -> B(x1:S)
-> Rules:
 a(a(x1:S)) -> a(d(d(d(x1:S))))
 a(b(x1:S)) -> b(c(a(x1:S)))
 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
 a(c(x1:S)) -> c(a(b(x1:S)))
 b(d(x1:S)) -> d(d(x1:S))
 b(c(x1:S)) -> c(b(b(x1:S)))
 d(a(x1:S)) -> d(d(c(x1:S)))
 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
 e(x1:S) -> a(x1:S)
->Projection:
 pi(B) = 1

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

Narrowing Processor:
-> Pairs:
 A(a(x1:S)) -> A(d(d(d(x1:S))))
 A(b(x1:S)) -> A(x1:S)
 A(d(d(c(x1:S)))) -> A(a(a(d(x1:S))))
 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
 A(d(d(c(x1:S)))) -> A(d(x1:S))
 A(c(x1:S)) -> A(b(x1:S))
-> Rules:
 a(a(x1:S)) -> a(d(d(d(x1:S))))
 a(b(x1:S)) -> b(c(a(x1:S)))
 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
 a(c(x1:S)) -> c(a(b(x1:S)))
 b(d(x1:S)) -> d(d(x1:S))
 b(c(x1:S)) -> c(b(b(x1:S)))
 d(a(x1:S)) -> d(d(c(x1:S)))
 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
 e(x1:S) -> a(x1:S)
->Narrowed Pairs:
->->Original Pair:
 A(a(x1:S)) -> A(d(d(d(x1:S))))
->-> Narrowed pairs:
 A(a(a(x1:S))) -> A(d(d(d(d(c(x1:S))))))

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

Subterm Processor:
-> Pairs:
 A(b(x1:S)) -> A(x1:S)
-> Rules:
 a(a(x1:S)) -> a(d(d(d(x1:S))))
 a(b(x1:S)) -> b(c(a(x1:S)))
 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
 a(c(x1:S)) -> c(a(b(x1:S)))
 b(d(x1:S)) -> d(d(x1:S))
 b(c(x1:S)) -> c(b(b(x1:S)))
 d(a(x1:S)) -> d(d(c(x1:S)))
 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
 e(x1:S) -> a(x1:S)
->Projection:
 pi(A) = 1

Problem 1.2: 

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

The problem is finite.

Problem 1.3: 

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

Problem 1.3: 

SCC Processor:
-> Pairs:
 E(e(f(f(x1:S)))) -> E(x1:S)
-> Rules:
 a(a(x1:S)) -> a(d(d(d(x1:S))))
 a(b(x1:S)) -> b(c(a(x1:S)))
 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
 a(c(x1:S)) -> c(a(b(x1:S)))
 b(d(x1:S)) -> d(d(x1:S))
 b(c(x1:S)) -> c(b(b(x1:S)))
 d(a(x1:S)) -> d(d(c(x1:S)))
 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
 e(x1:S) -> a(x1:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 E(e(f(f(x1:S)))) -> E(x1:S)
->->-> Rules:
 a(a(x1:S)) -> a(d(d(d(x1:S))))
 a(b(x1:S)) -> b(c(a(x1:S)))
 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
 a(c(x1:S)) -> c(a(b(x1:S)))
 b(d(x1:S)) -> d(d(x1:S))
 b(c(x1:S)) -> c(b(b(x1:S)))
 d(a(x1:S)) -> d(d(c(x1:S)))
 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
 e(x1:S) -> a(x1:S)

Problem 1.3: 

Subterm Processor:
-> Pairs:
 E(e(f(f(x1:S)))) -> E(x1:S)
-> Rules:
 a(a(x1:S)) -> a(d(d(d(x1:S))))
 a(b(x1:S)) -> b(c(a(x1:S)))
 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
 a(c(x1:S)) -> c(a(b(x1:S)))
 b(d(x1:S)) -> d(d(x1:S))
 b(c(x1:S)) -> c(b(b(x1:S)))
 d(a(x1:S)) -> d(d(c(x1:S)))
 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
 e(x1:S) -> a(x1:S)
->Projection:
 pi(E) = 1

Problem 1.3: 

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

The problem is finite.