0.00/0.19	YES
0.00/0.19	
0.00/0.19	Problem 1: 
0.00/0.19	
0.00/0.19	(VAR X Y)
0.00/0.19	(STRATEGY CONTEXTSENSITIVE
0.00/0.19	(diff 1 2)
0.00/0.19	(if 1)
0.00/0.19	(leq 1 2)
0.00/0.19	(p 1)
0.00/0.19	(0)
0.00/0.19	(false)
0.00/0.19	(s 1)
0.00/0.19	(true)
0.00/0.19	)
0.00/0.19	(RULES
0.00/0.19	diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	if(false,X,Y) -> Y
0.00/0.19	if(true,X,Y) -> X
0.00/0.19	leq(0,Y) -> true
0.00/0.19	leq(s(X),0) -> false
0.00/0.19	leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	p(0) -> 0
0.00/0.19	p(s(X)) -> X
0.00/0.19	)
0.00/0.19	
0.00/0.19	Problem 1: 
0.00/0.19	
0.00/0.19	Innermost Equivalent Processor:
0.00/0.19	-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	-> The context-sensitive term rewriting system is an orthogonal system. Therefore, innermost cs-termination implies cs-termination.
0.00/0.19	 
0.00/0.19	
0.00/0.19	Problem 1: 
0.00/0.19	
0.00/0.19	Dependency Pairs Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 DIFF(X,Y) -> IF(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 DIFF(X,Y) -> LEQ(X,Y)
0.00/0.19	 IF(false,X,Y) -> Y
0.00/0.19	 IF(true,X,Y) -> X
0.00/0.19	 LEQ(s(X),s(Y)) -> LEQ(X,Y)
0.00/0.19	-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	-> Unhiding Rules:
0.00/0.19	 s(diff(p(X),Y)) -> DIFF(p(X),Y)
0.00/0.19	 s(diff(p(X),Y)) -> P(X)
0.00/0.19	
0.00/0.19	Problem 1: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 DIFF(X,Y) -> IF(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 DIFF(X,Y) -> LEQ(X,Y)
0.00/0.19	 IF(false,X,Y) -> Y
0.00/0.19	 IF(true,X,Y) -> X
0.00/0.19	 LEQ(s(X),s(Y)) -> LEQ(X,Y)
0.00/0.19	-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	-> Unhiding rules:
0.00/0.19	 s(diff(p(X),Y)) -> DIFF(p(X),Y)
0.00/0.19	 s(diff(p(X),Y)) -> P(X)
0.00/0.19	->Strongly Connected Components:
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 LEQ(s(X),s(Y)) -> LEQ(X,Y)
0.00/0.19	->->-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	->->-> Unhiding rules:
0.00/0.19	 Empty
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 DIFF(X,Y) -> IF(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 IF(false,X,Y) -> Y
0.00/0.19	 IF(true,X,Y) -> X
0.00/0.19	->->-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	->->-> Unhiding rules:
0.00/0.19	 s(diff(p(X),Y)) -> DIFF(p(X),Y)
0.00/0.19	
0.00/0.19	
0.00/0.19	The problem is decomposed in 2 subproblems.
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	SubNColl Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 LEQ(s(X),s(Y)) -> LEQ(X,Y)
0.00/0.19	-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	-> Unhiding rules:
0.00/0.19	 Empty
0.00/0.19	->Projection:
0.00/0.19	 pi(LEQ) = 1
0.00/0.19	
0.00/0.19	Problem 1.1: 
0.00/0.19	
0.00/0.19	Basic Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 Empty
0.00/0.19	-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	-> Unhiding rules:
0.00/0.19	 Empty
0.00/0.19	-> Result:
0.00/0.19	 Set P is empty
0.00/0.19	
0.00/0.19	The problem is finite.
0.00/0.19	
0.00/0.19	Problem 1.2: 
0.00/0.19	
0.00/0.19	Reduction Pairs Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 DIFF(X,Y) -> IF(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 IF(false,X,Y) -> Y
0.00/0.19	 IF(true,X,Y) -> X
0.00/0.19	-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	-> Unhiding rules:
0.00/0.19	 s(diff(p(X),Y)) -> DIFF(p(X),Y)
0.00/0.19	-> Usable rules:
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	->Interpretation type:
0.00/0.19	 Linear
0.00/0.19	->Coefficients:
0.00/0.19	 All rationals
0.00/0.19	->Dimension:
0.00/0.19	 1
0.00/0.19	->Bound:
0.00/0.19	 2
0.00/0.19	->Interpretation:
0.00/0.19	 
0.00/0.19	[diff](X1,X2) = X1 + X2 + 1/2
0.00/0.19	[leq](X1,X2) = X1
0.00/0.19	[p](X) = 1/2.X
0.00/0.19	[0] = 0
0.00/0.19	[false] = 1/2
0.00/0.19	[s](X) = 2.X + 1
0.00/0.19	[true] = 0
0.00/0.19	[DIFF](X1,X2) = 2.X1 + 2.X2 + 2
0.00/0.19	[IF](X1,X2,X3) = 1/2.X1 + 2.X2 + X3
0.00/0.19	
0.00/0.19	Problem 1.2: 
0.00/0.19	
0.00/0.19	SCC Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 DIFF(X,Y) -> IF(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 IF(true,X,Y) -> X
0.00/0.19	-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	-> Unhiding rules:
0.00/0.19	 s(diff(p(X),Y)) -> DIFF(p(X),Y)
0.00/0.19	->Strongly Connected Components:
0.00/0.19	->->Cycle:
0.00/0.19	->->-> Pairs:
0.00/0.19	 DIFF(X,Y) -> IF(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 IF(true,X,Y) -> X
0.00/0.19	->->-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	->->-> Unhiding rules:
0.00/0.19	 s(diff(p(X),Y)) -> DIFF(p(X),Y)
0.00/0.19	
0.00/0.19	Problem 1.2: 
0.00/0.19	
0.00/0.19	Reduction Pairs Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 DIFF(X,Y) -> IF(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 IF(true,X,Y) -> X
0.00/0.19	-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	-> Unhiding rules:
0.00/0.19	 s(diff(p(X),Y)) -> DIFF(p(X),Y)
0.00/0.19	-> Usable rules:
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	->Interpretation type:
0.00/0.19	 Linear
0.00/0.19	->Coefficients:
0.00/0.19	 Natural Numbers
0.00/0.19	->Dimension:
0.00/0.19	 1
0.00/0.19	->Bound:
0.00/0.19	 2
0.00/0.19	->Interpretation:
0.00/0.19	 
0.00/0.19	[diff](X1,X2) = 2.X1 + 2.X2
0.00/0.19	[leq](X1,X2) = 2.X1
0.00/0.19	[p](X) = 2.X + 2
0.00/0.19	[0] = 0
0.00/0.19	[false] = 0
0.00/0.19	[s](X) = 2.X
0.00/0.19	[true] = 0
0.00/0.19	[DIFF](X1,X2) = 2.X1 + 2
0.00/0.19	[IF](X1,X2,X3) = X1 + X2 + 1
0.00/0.19	
0.00/0.19	Problem 1.2: 
0.00/0.19	
0.00/0.19	Basic Processor:
0.00/0.19	-> Pairs:
0.00/0.19	 IF(true,X,Y) -> X
0.00/0.19	-> Rules:
0.00/0.19	 diff(X,Y) -> if(leq(X,Y),0,s(diff(p(X),Y)))
0.00/0.19	 if(false,X,Y) -> Y
0.00/0.19	 if(true,X,Y) -> X
0.00/0.19	 leq(0,Y) -> true
0.00/0.19	 leq(s(X),0) -> false
0.00/0.19	 leq(s(X),s(Y)) -> leq(X,Y)
0.00/0.19	 p(0) -> 0
0.00/0.19	 p(s(X)) -> X
0.00/0.19	-> Unhiding rules:
0.00/0.19	 s(diff(p(X),Y)) -> DIFF(p(X),Y)
0.00/0.19	-> Result:
0.00/0.19	 All pairs P are from Px1
0.00/0.19	
0.00/0.19	The problem is finite.
0.00/0.20	EOF