145.80/146.18	YES
145.80/146.18	
145.80/146.18	Problem 1: 
145.80/146.18	
145.80/146.18	(VAR v_NonEmpty:S x1:S)
145.80/146.18	(RULES
145.80/146.18	a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	e(x1:S) -> a(x1:S)
145.80/146.18	)
145.80/146.18	
145.80/146.18	Problem 1: 
145.80/146.18	
145.80/146.18	Dependency Pairs Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(a(x1:S)) -> A(d(d(d(x1:S))))
145.80/146.18	 A(a(x1:S)) -> D(d(d(x1:S)))
145.80/146.18	 A(a(x1:S)) -> D(d(x1:S))
145.80/146.18	 A(a(x1:S)) -> D(x1:S)
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(b(x1:S)) -> B(c(a(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(a(d(x1:S))))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(d(d(c(x1:S)))) -> D(x1:S)
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	 A(c(x1:S)) -> B(x1:S)
145.80/146.18	 B(d(x1:S)) -> D(d(x1:S))
145.80/146.18	 B(c(x1:S)) -> B(b(x1:S))
145.80/146.18	 B(c(x1:S)) -> B(x1:S)
145.80/146.18	 E(e(f(f(x1:S)))) -> E(e(x1:S))
145.80/146.18	 E(e(f(f(x1:S)))) -> E(x1:S)
145.80/146.18	 E(x1:S) -> A(x1:S)
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	
145.80/146.18	Problem 1: 
145.80/146.18	
145.80/146.18	SCC Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(a(x1:S)) -> A(d(d(d(x1:S))))
145.80/146.18	 A(a(x1:S)) -> D(d(d(x1:S)))
145.80/146.18	 A(a(x1:S)) -> D(d(x1:S))
145.80/146.18	 A(a(x1:S)) -> D(x1:S)
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(b(x1:S)) -> B(c(a(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(a(d(x1:S))))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(d(d(c(x1:S)))) -> D(x1:S)
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	 A(c(x1:S)) -> B(x1:S)
145.80/146.18	 B(d(x1:S)) -> D(d(x1:S))
145.80/146.18	 B(c(x1:S)) -> B(b(x1:S))
145.80/146.18	 B(c(x1:S)) -> B(x1:S)
145.80/146.18	 E(e(f(f(x1:S)))) -> E(e(x1:S))
145.80/146.18	 E(e(f(f(x1:S)))) -> E(x1:S)
145.80/146.18	 E(x1:S) -> A(x1:S)
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Strongly Connected Components:
145.80/146.18	->->Cycle:
145.80/146.18	->->-> Pairs:
145.80/146.18	 B(c(x1:S)) -> B(b(x1:S))
145.80/146.18	 B(c(x1:S)) -> B(x1:S)
145.80/146.18	->->-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->->Cycle:
145.80/146.18	->->-> Pairs:
145.80/146.18	 A(a(x1:S)) -> A(d(d(d(x1:S))))
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(a(d(x1:S))))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	->->-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->->Cycle:
145.80/146.18	->->-> Pairs:
145.80/146.18	 E(e(f(f(x1:S)))) -> E(e(x1:S))
145.80/146.18	 E(e(f(f(x1:S)))) -> E(x1:S)
145.80/146.18	->->-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	
145.80/146.18	
145.80/146.18	The problem is decomposed in 3 subproblems.
145.80/146.18	
145.80/146.18	Problem 1.1: 
145.80/146.18	
145.80/146.18	Reduction Pair Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 B(c(x1:S)) -> B(b(x1:S))
145.80/146.18	 B(c(x1:S)) -> B(x1:S)
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	-> Usable rules:
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	->Interpretation type:
145.80/146.18	 Linear
145.80/146.18	->Coefficients:
145.80/146.18	 Natural Numbers
145.80/146.18	->Dimension:
145.80/146.18	 1
145.80/146.18	->Bound:
145.80/146.18	 2
145.80/146.18	->Interpretation:
145.80/146.18	 
145.80/146.18	[a](X) = 2.X + 2
145.80/146.18	[b](X) = X
145.80/146.18	[d](X) = X
145.80/146.18	[c](X) = 2.X + 2
145.80/146.18	[B](X) = X
145.80/146.18	
145.80/146.18	Problem 1.1: 
145.80/146.18	
145.80/146.18	SCC Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 B(c(x1:S)) -> B(x1:S)
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Strongly Connected Components:
145.80/146.18	->->Cycle:
145.80/146.18	->->-> Pairs:
145.80/146.18	 B(c(x1:S)) -> B(x1:S)
145.80/146.18	->->-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	
145.80/146.18	Problem 1.1: 
145.80/146.18	
145.80/146.18	Subterm Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 B(c(x1:S)) -> B(x1:S)
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Projection:
145.80/146.18	 pi(B) = 1
145.80/146.18	
145.80/146.18	Problem 1.1: 
145.80/146.18	
145.80/146.18	SCC Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 Empty
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Strongly Connected Components:
145.80/146.18	 There is no strongly connected component
145.80/146.18	
145.80/146.18	The problem is finite.
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	Narrowing Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(a(x1:S)) -> A(d(d(d(x1:S))))
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(a(d(x1:S))))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Narrowed Pairs:
145.80/146.18	->->Original Pair:
145.80/146.18	 A(a(x1:S)) -> A(d(d(d(x1:S))))
145.80/146.18	->-> Narrowed pairs:
145.80/146.18	 A(a(a(x1:S))) -> A(d(d(d(d(c(x1:S))))))
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	SCC Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(a(a(x1:S))) -> A(d(d(d(d(c(x1:S))))))
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(a(d(x1:S))))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Strongly Connected Components:
145.80/146.18	->->Cycle:
145.80/146.18	->->-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(a(d(x1:S))))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	->->-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	Reduction Pair Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(a(d(x1:S))))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	-> Usable rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	->Interpretation type:
145.80/146.18	 Linear
145.80/146.18	->Coefficients:
145.80/146.18	 Natural Numbers
145.80/146.18	->Dimension:
145.80/146.18	 1
145.80/146.18	->Bound:
145.80/146.18	 2
145.80/146.18	->Interpretation:
145.80/146.18	 
145.80/146.18	[a](X) = 1
145.80/146.18	[b](X) = X
145.80/146.18	[d](X) = 2
145.80/146.18	[c](X) = X
145.80/146.18	[A](X) = X
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	SCC Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Strongly Connected Components:
145.80/146.18	->->Cycle:
145.80/146.18	->->-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	->->-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	Reduction Pair Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(a(d(x1:S)))
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	-> Usable rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	->Interpretation type:
145.80/146.18	 Linear
145.80/146.18	->Coefficients:
145.80/146.18	 Natural Numbers
145.80/146.18	->Dimension:
145.80/146.18	 1
145.80/146.18	->Bound:
145.80/146.18	 2
145.80/146.18	->Interpretation:
145.80/146.18	 
145.80/146.18	[a](X) = 0
145.80/146.18	[b](X) = X
145.80/146.18	[d](X) = 2
145.80/146.18	[c](X) = 2.X
145.80/146.18	[A](X) = 2.X
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	SCC Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Strongly Connected Components:
145.80/146.18	->->Cycle:
145.80/146.18	->->-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	->->-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	Reduction Pair Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(d(d(c(x1:S)))) -> A(d(x1:S))
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	-> Usable rules:
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	->Interpretation type:
145.80/146.18	 Linear
145.80/146.18	->Coefficients:
145.80/146.18	 Natural Numbers
145.80/146.18	->Dimension:
145.80/146.18	 1
145.80/146.18	->Bound:
145.80/146.18	 2
145.80/146.18	->Interpretation:
145.80/146.18	 
145.80/146.18	[a](X) = 2.X + 2
145.80/146.18	[b](X) = X
145.80/146.18	[d](X) = X
145.80/146.18	[c](X) = 2.X + 1
145.80/146.18	[A](X) = 2.X
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	SCC Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Strongly Connected Components:
145.80/146.18	->->Cycle:
145.80/146.18	->->-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	->->-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	Reduction Pair Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	 A(c(x1:S)) -> A(b(x1:S))
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	-> Usable rules:
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	->Interpretation type:
145.80/146.18	 Linear
145.80/146.18	->Coefficients:
145.80/146.18	 Natural Numbers
145.80/146.18	->Dimension:
145.80/146.18	 1
145.80/146.18	->Bound:
145.80/146.18	 2
145.80/146.18	->Interpretation:
145.80/146.18	 
145.80/146.18	[a](X) = 2.X + 1
145.80/146.18	[b](X) = X
145.80/146.18	[d](X) = X
145.80/146.18	[c](X) = 2.X + 1
145.80/146.18	[A](X) = 2.X
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	SCC Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Strongly Connected Components:
145.80/146.18	->->Cycle:
145.80/146.18	->->-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	->->-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	Subterm Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 A(b(x1:S)) -> A(x1:S)
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Projection:
145.80/146.18	 pi(A) = 1
145.80/146.18	
145.80/146.18	Problem 1.2: 
145.80/146.18	
145.80/146.18	SCC Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 Empty
145.80/146.18	-> Rules:
145.80/146.18	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.18	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.18	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.18	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.18	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.18	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.18	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.18	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.18	 e(x1:S) -> a(x1:S)
145.80/146.18	->Strongly Connected Components:
145.80/146.18	 There is no strongly connected component
145.80/146.18	
145.80/146.18	The problem is finite.
145.80/146.18	
145.80/146.18	Problem 1.3: 
145.80/146.18	
145.80/146.18	Reduction Pair Processor:
145.80/146.18	-> Pairs:
145.80/146.18	 E(e(f(f(x1:S)))) -> E(e(x1:S))
145.80/146.19	 E(e(f(f(x1:S)))) -> E(x1:S)
145.80/146.19	-> Rules:
145.80/146.19	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.19	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.19	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.19	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.19	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.19	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.19	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.19	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.19	 e(x1:S) -> a(x1:S)
145.80/146.19	-> Usable rules:
145.80/146.19	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.19	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.19	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.19	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.19	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.19	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.19	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.19	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.19	 e(x1:S) -> a(x1:S)
145.80/146.19	->Interpretation type:
145.80/146.19	 Linear
145.80/146.19	->Coefficients:
145.80/146.19	 Natural Numbers
145.80/146.19	->Dimension:
145.80/146.19	 1
145.80/146.19	->Bound:
145.80/146.19	 2
145.80/146.19	->Interpretation:
145.80/146.19	 
145.80/146.19	[a](X) = 0
145.80/146.19	[b](X) = 2.X
145.80/146.19	[d](X) = 2.X
145.80/146.19	[e](X) = 2.X + 2
145.80/146.19	[c](X) = 0
145.80/146.19	[f](X) = X + 2
145.80/146.19	[E](X) = X
145.80/146.19	
145.80/146.19	Problem 1.3: 
145.80/146.19	
145.80/146.19	SCC Processor:
145.80/146.19	-> Pairs:
145.80/146.19	 E(e(f(f(x1:S)))) -> E(x1:S)
145.80/146.19	-> Rules:
145.80/146.19	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.19	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.19	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.19	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.19	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.19	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.19	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.19	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.19	 e(x1:S) -> a(x1:S)
145.80/146.19	->Strongly Connected Components:
145.80/146.19	->->Cycle:
145.80/146.19	->->-> Pairs:
145.80/146.19	 E(e(f(f(x1:S)))) -> E(x1:S)
145.80/146.19	->->-> Rules:
145.80/146.19	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.19	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.19	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.19	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.19	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.19	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.19	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.19	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.19	 e(x1:S) -> a(x1:S)
145.80/146.19	
145.80/146.19	Problem 1.3: 
145.80/146.19	
145.80/146.19	Subterm Processor:
145.80/146.19	-> Pairs:
145.80/146.19	 E(e(f(f(x1:S)))) -> E(x1:S)
145.80/146.19	-> Rules:
145.80/146.19	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.19	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.19	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.19	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.19	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.19	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.19	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.19	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.19	 e(x1:S) -> a(x1:S)
145.80/146.19	->Projection:
145.80/146.19	 pi(E) = 1
145.80/146.19	
145.80/146.19	Problem 1.3: 
145.80/146.19	
145.80/146.19	SCC Processor:
145.80/146.19	-> Pairs:
145.80/146.19	 Empty
145.80/146.19	-> Rules:
145.80/146.19	 a(a(x1:S)) -> a(d(d(d(x1:S))))
145.80/146.19	 a(b(x1:S)) -> b(c(a(x1:S)))
145.80/146.19	 a(d(d(c(x1:S)))) -> a(a(a(d(x1:S))))
145.80/146.19	 a(c(x1:S)) -> c(a(b(x1:S)))
145.80/146.19	 b(d(x1:S)) -> d(d(x1:S))
145.80/146.19	 b(c(x1:S)) -> c(b(b(x1:S)))
145.80/146.19	 d(a(x1:S)) -> d(d(c(x1:S)))
145.80/146.19	 e(e(f(f(x1:S)))) -> f(f(f(e(e(x1:S)))))
145.80/146.19	 e(x1:S) -> a(x1:S)
145.80/146.19	->Strongly Connected Components:
145.80/146.19	 There is no strongly connected component
145.80/146.19	
145.80/146.19	The problem is finite.
145.80/146.19	EOF