/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

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

Problem 1: 

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

Problem 1: 

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


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 B(c(x1)) -> B(b(x1))
 B(c(x1)) -> B(x1)
-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)
-> Usable rules:
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
->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) = 2.X

Problem 1.1: 

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

Problem 1.1: 

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

Problem 1.1: 

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

The problem is finite.

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 A(b(x1)) -> A(x1)
 A(d(d(c(x1)))) -> A(a(a(d(x1))))
 A(d(d(c(x1)))) -> A(a(d(x1)))
 A(d(d(c(x1)))) -> A(d(x1))
 A(c(x1)) -> A(b(x1))
-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)
-> Usable rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
->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)) -> A(x1)
 A(d(d(c(x1)))) -> A(a(d(x1)))
 A(d(d(c(x1)))) -> A(d(x1))
 A(c(x1)) -> A(b(x1))
-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(b(x1)) -> A(x1)
 A(d(d(c(x1)))) -> A(a(d(x1)))
 A(d(d(c(x1)))) -> A(d(x1))
 A(c(x1)) -> A(b(x1))
->->-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 A(b(x1)) -> A(x1)
 A(d(d(c(x1)))) -> A(a(d(x1)))
 A(d(d(c(x1)))) -> A(d(x1))
 A(c(x1)) -> A(b(x1))
-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)
-> Usable rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
->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)) -> A(x1)
 A(d(d(c(x1)))) -> A(d(x1))
 A(c(x1)) -> A(b(x1))
-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(b(x1)) -> A(x1)
 A(d(d(c(x1)))) -> A(d(x1))
 A(c(x1)) -> A(b(x1))
->->-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 A(b(x1)) -> A(x1)
 A(d(d(c(x1)))) -> A(d(x1))
 A(c(x1)) -> A(b(x1))
-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)
-> Usable rules:
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
->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
[A](X) = 2.X

Problem 1.2: 

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

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 A(b(x1)) -> A(x1)
 A(c(x1)) -> A(b(x1))
-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)
-> Usable rules:
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
->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)) -> A(x1)
-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A(b(x1)) -> A(x1)
->->-> Rules:
 a(a(x1)) -> a(d(d(d(x1))))
 a(b(x1)) -> b(c(a(x1)))
 a(d(d(c(x1)))) -> a(a(a(d(x1))))
 a(c(x1)) -> c(a(b(x1)))
 b(d(x1)) -> d(d(x1))
 b(c(x1)) -> c(b(b(x1)))
 d(a(x1)) -> d(d(c(x1)))
 e(e(f(f(x1)))) -> f(f(f(e(e(x1)))))
 e(x1) -> a(x1)

Problem 1.2: 

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

Problem 1.2: 

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

Problem 1.3: 

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

Problem 1.3: 

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

Problem 1.3: 

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

The problem is finite.