4.22/4.69	YES
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	(VAR v_NonEmpty:S x1:S)
4.22/4.69	(RULES
4.22/4.69	a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	c(c(x1:S)) -> b(x1:S)
4.22/4.69	c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	)
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	Dependency Pairs Processor:
4.22/4.69	-> Pairs:
4.22/4.69	 A(a(x1:S)) -> C(x1:S)
4.22/4.69	 A(a(x1:S)) -> D(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	 D(d(x1:S)) -> A(c(x1:S))
4.22/4.69	 D(d(x1:S)) -> B(a(c(x1:S)))
4.22/4.69	 D(d(x1:S)) -> C(x1:S)
4.22/4.69	-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	SCC Processor:
4.22/4.69	-> Pairs:
4.22/4.69	 A(a(x1:S)) -> C(x1:S)
4.22/4.69	 A(a(x1:S)) -> D(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	 D(d(x1:S)) -> A(c(x1:S))
4.22/4.69	 D(d(x1:S)) -> B(a(c(x1:S)))
4.22/4.69	 D(d(x1:S)) -> C(x1:S)
4.22/4.69	-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	->Strongly Connected Components:
4.22/4.69	->->Cycle:
4.22/4.69	->->-> Pairs:
4.22/4.69	 A(a(x1:S)) -> C(x1:S)
4.22/4.69	 A(a(x1:S)) -> D(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	 D(d(x1:S)) -> A(c(x1:S))
4.22/4.69	 D(d(x1:S)) -> B(a(c(x1:S)))
4.22/4.69	 D(d(x1:S)) -> C(x1:S)
4.22/4.69	->->-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	Reduction Pair Processor:
4.22/4.69	-> Pairs:
4.22/4.69	 A(a(x1:S)) -> C(x1:S)
4.22/4.69	 A(a(x1:S)) -> D(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	 D(d(x1:S)) -> A(c(x1:S))
4.22/4.69	 D(d(x1:S)) -> B(a(c(x1:S)))
4.22/4.69	 D(d(x1:S)) -> C(x1:S)
4.22/4.69	-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	-> Usable rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	->Interpretation type:
4.22/4.69	 Linear
4.22/4.69	->Coefficients:
4.22/4.69	 All rationals
4.22/4.69	->Dimension:
4.22/4.69	 1
4.22/4.69	->Bound:
4.22/4.69	 4
4.22/4.69	->Interpretation:
4.22/4.69	 
4.22/4.69	[a](X) = X + 1
4.22/4.69	[b](X) = X + 1
4.22/4.69	[c](X) = X + 1/2
4.22/4.69	[d](X) = X + 3/2
4.22/4.69	[A](X) = 1/4.X + 3/4
4.22/4.69	[B](X) = 1/4.X + 3/4
4.22/4.69	[C](X) = 1/4.X + 3/4
4.22/4.69	[D](X) = 1/4.X + 3/4
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	SCC Processor:
4.22/4.69	-> Pairs:
4.22/4.69	 A(a(x1:S)) -> D(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	 D(d(x1:S)) -> A(c(x1:S))
4.22/4.69	 D(d(x1:S)) -> B(a(c(x1:S)))
4.22/4.69	 D(d(x1:S)) -> C(x1:S)
4.22/4.69	-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	->Strongly Connected Components:
4.22/4.69	->->Cycle:
4.22/4.69	->->-> Pairs:
4.22/4.69	 A(a(x1:S)) -> D(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	 D(d(x1:S)) -> A(c(x1:S))
4.22/4.69	 D(d(x1:S)) -> B(a(c(x1:S)))
4.22/4.69	 D(d(x1:S)) -> C(x1:S)
4.22/4.69	->->-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	Reduction Pair Processor:
4.22/4.69	-> Pairs:
4.22/4.69	 A(a(x1:S)) -> D(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	 D(d(x1:S)) -> A(c(x1:S))
4.22/4.69	 D(d(x1:S)) -> B(a(c(x1:S)))
4.22/4.69	 D(d(x1:S)) -> C(x1:S)
4.22/4.69	-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	-> Usable rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	->Interpretation type:
4.22/4.69	 Linear
4.22/4.69	->Coefficients:
4.22/4.69	 All rationals
4.22/4.69	->Dimension:
4.22/4.69	 1
4.22/4.69	->Bound:
4.22/4.69	 4
4.22/4.69	->Interpretation:
4.22/4.69	 
4.22/4.69	[a](X) = X + 2/3
4.22/4.69	[b](X) = X + 2/3
4.22/4.69	[c](X) = X + 1/3
4.22/4.69	[d](X) = X + 1
4.22/4.69	[A](X) = 2.X + 4/3
4.22/4.69	[B](X) = 2.X + 4/3
4.22/4.69	[C](X) = 2.X + 2/3
4.22/4.69	[D](X) = 2.X + 4/3
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	SCC Processor:
4.22/4.69	-> Pairs:
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	 D(d(x1:S)) -> A(c(x1:S))
4.22/4.69	 D(d(x1:S)) -> B(a(c(x1:S)))
4.22/4.69	 D(d(x1:S)) -> C(x1:S)
4.22/4.69	-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	->Strongly Connected Components:
4.22/4.69	->->Cycle:
4.22/4.69	->->-> Pairs:
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	->->-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	Reduction Pair Processor:
4.22/4.69	-> Pairs:
4.22/4.69	 A(b(x1:S)) -> C(c(c(x1:S)))
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	-> Usable rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	->Interpretation type:
4.22/4.69	 Linear
4.22/4.69	->Coefficients:
4.22/4.69	 All rationals
4.22/4.69	->Dimension:
4.22/4.69	 1
4.22/4.69	->Bound:
4.22/4.69	 4
4.22/4.69	->Interpretation:
4.22/4.69	 
4.22/4.69	[a](X) = X + 4/3
4.22/4.69	[b](X) = X + 4/3
4.22/4.69	[c](X) = X + 2/3
4.22/4.69	[d](X) = X + 2
4.22/4.69	[A](X) = 2.X + 1/4
4.22/4.69	[B](X) = 2.X + 1/3
4.22/4.69	[C](X) = 2.X
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	SCC Processor:
4.22/4.69	-> Pairs:
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	->Strongly Connected Components:
4.22/4.69	->->Cycle:
4.22/4.69	->->-> Pairs:
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	->->-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.69	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.69	
4.22/4.69	Problem 1: 
4.22/4.69	
4.22/4.69	Reduction Pair Processor:
4.22/4.69	-> Pairs:
4.22/4.69	 A(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 A(b(x1:S)) -> C(x1:S)
4.22/4.69	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.69	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.69	 B(b(x1:S)) -> C(x1:S)
4.22/4.69	 C(c(x1:S)) -> B(x1:S)
4.22/4.69	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.69	 C(d(x1:S)) -> A(x1:S)
4.22/4.69	-> Rules:
4.22/4.69	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.69	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.69	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.69	 c(c(x1:S)) -> b(x1:S)
4.22/4.69	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	-> Usable rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	->Interpretation type:
4.22/4.70	 Linear
4.22/4.70	->Coefficients:
4.22/4.70	 All rationals
4.22/4.70	->Dimension:
4.22/4.70	 1
4.22/4.70	->Bound:
4.22/4.70	 4
4.22/4.70	->Interpretation:
4.22/4.70	 
4.22/4.70	[a](X) = X + 2
4.22/4.70	[b](X) = X + 2
4.22/4.70	[c](X) = X + 1
4.22/4.70	[d](X) = X + 3
4.22/4.70	[A](X) = 2.X
4.22/4.70	[B](X) = 2.X + 1/2
4.22/4.70	[C](X) = 2.X + 2/3
4.22/4.70	
4.22/4.70	Problem 1: 
4.22/4.70	
4.22/4.70	SCC Processor:
4.22/4.70	-> Pairs:
4.22/4.70	 A(b(x1:S)) -> C(x1:S)
4.22/4.70	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.70	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.70	 B(b(x1:S)) -> C(x1:S)
4.22/4.70	 C(c(x1:S)) -> B(x1:S)
4.22/4.70	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.70	 C(d(x1:S)) -> A(x1:S)
4.22/4.70	-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	->Strongly Connected Components:
4.22/4.70	->->Cycle:
4.22/4.70	->->-> Pairs:
4.22/4.70	 A(b(x1:S)) -> C(x1:S)
4.22/4.70	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.70	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.70	 B(b(x1:S)) -> C(x1:S)
4.22/4.70	 C(c(x1:S)) -> B(x1:S)
4.22/4.70	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.70	 C(d(x1:S)) -> A(x1:S)
4.22/4.70	->->-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	
4.22/4.70	Problem 1: 
4.22/4.70	
4.22/4.70	Reduction Pair Processor:
4.22/4.70	-> Pairs:
4.22/4.70	 A(b(x1:S)) -> C(x1:S)
4.22/4.70	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.70	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.70	 B(b(x1:S)) -> C(x1:S)
4.22/4.70	 C(c(x1:S)) -> B(x1:S)
4.22/4.70	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.70	 C(d(x1:S)) -> A(x1:S)
4.22/4.70	-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	-> Usable rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	->Interpretation type:
4.22/4.70	 Linear
4.22/4.70	->Coefficients:
4.22/4.70	 All rationals
4.22/4.70	->Dimension:
4.22/4.70	 1
4.22/4.70	->Bound:
4.22/4.70	 4
4.22/4.70	->Interpretation:
4.22/4.70	 
4.22/4.70	[a](X) = X + 2
4.22/4.70	[b](X) = X + 2
4.22/4.70	[c](X) = X + 1
4.22/4.70	[d](X) = X + 3
4.22/4.70	[A](X) = 1/4.X + 2
4.22/4.70	[B](X) = 1/4.X + 2
4.22/4.70	[C](X) = 1/4.X + 2
4.22/4.70	
4.22/4.70	Problem 1: 
4.22/4.70	
4.22/4.70	SCC Processor:
4.22/4.70	-> Pairs:
4.22/4.70	 B(b(x1:S)) -> A(c(c(x1:S)))
4.22/4.70	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.70	 B(b(x1:S)) -> C(x1:S)
4.22/4.70	 C(c(x1:S)) -> B(x1:S)
4.22/4.70	 C(d(x1:S)) -> A(a(x1:S))
4.22/4.70	 C(d(x1:S)) -> A(x1:S)
4.22/4.70	-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	->Strongly Connected Components:
4.22/4.70	->->Cycle:
4.22/4.70	->->-> Pairs:
4.22/4.70	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.70	 B(b(x1:S)) -> C(x1:S)
4.22/4.70	 C(c(x1:S)) -> B(x1:S)
4.22/4.70	->->-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	
4.22/4.70	Problem 1: 
4.22/4.70	
4.22/4.70	Reduction Pair Processor:
4.22/4.70	-> Pairs:
4.22/4.70	 B(b(x1:S)) -> C(c(x1:S))
4.22/4.70	 B(b(x1:S)) -> C(x1:S)
4.22/4.70	 C(c(x1:S)) -> B(x1:S)
4.22/4.70	-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	-> Usable rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	->Interpretation type:
4.22/4.70	 Linear
4.22/4.70	->Coefficients:
4.22/4.70	 All rationals
4.22/4.70	->Dimension:
4.22/4.70	 1
4.22/4.70	->Bound:
4.22/4.70	 4
4.22/4.70	->Interpretation:
4.22/4.70	 
4.22/4.70	[a](X) = X + 4/3
4.22/4.70	[b](X) = X + 4/3
4.22/4.70	[c](X) = X + 2/3
4.22/4.70	[d](X) = X + 2
4.22/4.70	[B](X) = 2.X + 1/4
4.22/4.70	[C](X) = 2.X + 2/3
4.22/4.70	
4.22/4.70	Problem 1: 
4.22/4.70	
4.22/4.70	SCC Processor:
4.22/4.70	-> Pairs:
4.22/4.70	 B(b(x1:S)) -> C(x1:S)
4.22/4.70	 C(c(x1:S)) -> B(x1:S)
4.22/4.70	-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	->Strongly Connected Components:
4.22/4.70	->->Cycle:
4.22/4.70	->->-> Pairs:
4.22/4.70	 B(b(x1:S)) -> C(x1:S)
4.22/4.70	 C(c(x1:S)) -> B(x1:S)
4.22/4.70	->->-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	
4.22/4.70	Problem 1: 
4.22/4.70	
4.22/4.70	Subterm Processor:
4.22/4.70	-> Pairs:
4.22/4.70	 B(b(x1:S)) -> C(x1:S)
4.22/4.70	 C(c(x1:S)) -> B(x1:S)
4.22/4.70	-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	->Projection:
4.22/4.70	 pi(B) = 1
4.22/4.70	 pi(C) = 1
4.22/4.70	
4.22/4.70	Problem 1: 
4.22/4.70	
4.22/4.70	SCC Processor:
4.22/4.70	-> Pairs:
4.22/4.70	 Empty
4.22/4.70	-> Rules:
4.22/4.70	 a(a(x1:S)) -> d(c(x1:S))
4.22/4.70	 a(b(x1:S)) -> c(c(c(x1:S)))
4.22/4.70	 b(b(x1:S)) -> a(c(c(x1:S)))
4.22/4.70	 c(c(x1:S)) -> b(x1:S)
4.22/4.70	 c(d(x1:S)) -> a(a(x1:S))
4.22/4.70	 d(d(x1:S)) -> b(a(c(x1:S)))
4.22/4.70	->Strongly Connected Components:
4.22/4.70	 There is no strongly connected component
4.22/4.70	
4.22/4.70	The problem is finite.
4.22/4.70	EOF