0.00/0.21	YES
0.00/0.21	
0.00/0.21	Problem 1: 
0.00/0.21	
0.00/0.21	(VAR X Y)
0.00/0.21	(STRATEGY CONTEXTSENSITIVE
0.00/0.21	(div 1 2)
0.00/0.21	(gt 1 2)
0.00/0.21	(if 1)
0.00/0.21	(minus 1 2)
0.00/0.21	(p 1)
0.00/0.21	(0)
0.00/0.21	(false)
0.00/0.21	(s 1)
0.00/0.21	(true)
0.00/0.21	)
0.00/0.21	(RULES
0.00/0.21	div(0,s(Y)) -> 0
0.00/0.21	div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	gt(0,Y) -> false
0.00/0.21	gt(s(X),0) -> true
0.00/0.21	gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	if(false,X,Y) -> Y
0.00/0.21	if(true,X,Y) -> X
0.00/0.21	minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	p(0) -> 0
0.00/0.21	p(s(X)) -> X
0.00/0.21	)
0.00/0.21	
0.00/0.21	Problem 1: 
0.00/0.21	
0.00/0.21	Innermost Equivalent Processor:
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> The context-sensitive term rewriting system is an orthogonal system. Therefore, innermost cs-termination implies cs-termination.
0.00/0.21	 
0.00/0.21	
0.00/0.21	Problem 1: 
0.00/0.21	
0.00/0.21	Dependency Pairs Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 DIV(s(X),s(Y)) -> DIV(minus(X,Y),s(Y))
0.00/0.21	 DIV(s(X),s(Y)) -> MINUS(X,Y)
0.00/0.21	 GT(s(X),s(Y)) -> GT(X,Y)
0.00/0.21	 IF(false,X,Y) -> Y
0.00/0.21	 IF(true,X,Y) -> X
0.00/0.21	 MINUS(X,Y) -> GT(Y,0)
0.00/0.21	 MINUS(X,Y) -> IF(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding Rules:
0.00/0.21	 minus(p(X),p(Y)) -> MINUS(p(X),p(Y))
0.00/0.21	 minus(p(X),p(Y)) -> P(X)
0.00/0.21	 minus(p(X),p(Y)) -> P(Y)
0.00/0.21	
0.00/0.21	Problem 1: 
0.00/0.21	
0.00/0.21	SCC Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 DIV(s(X),s(Y)) -> DIV(minus(X,Y),s(Y))
0.00/0.21	 DIV(s(X),s(Y)) -> MINUS(X,Y)
0.00/0.21	 GT(s(X),s(Y)) -> GT(X,Y)
0.00/0.21	 IF(false,X,Y) -> Y
0.00/0.21	 IF(true,X,Y) -> X
0.00/0.21	 MINUS(X,Y) -> GT(Y,0)
0.00/0.21	 MINUS(X,Y) -> IF(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding rules:
0.00/0.21	 minus(p(X),p(Y)) -> MINUS(p(X),p(Y))
0.00/0.21	 minus(p(X),p(Y)) -> P(X)
0.00/0.21	 minus(p(X),p(Y)) -> P(Y)
0.00/0.21	->Strongly Connected Components:
0.00/0.21	->->Cycle:
0.00/0.21	->->-> Pairs:
0.00/0.21	 IF(false,X,Y) -> Y
0.00/0.21	 IF(true,X,Y) -> X
0.00/0.21	 MINUS(X,Y) -> IF(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	->->-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	->->-> Unhiding rules:
0.00/0.21	 minus(p(X),p(Y)) -> MINUS(p(X),p(Y))
0.00/0.21	->->Cycle:
0.00/0.21	->->-> Pairs:
0.00/0.21	 GT(s(X),s(Y)) -> GT(X,Y)
0.00/0.21	->->-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	->->-> Unhiding rules:
0.00/0.21	 Empty
0.00/0.21	->->Cycle:
0.00/0.21	->->-> Pairs:
0.00/0.21	 DIV(s(X),s(Y)) -> DIV(minus(X,Y),s(Y))
0.00/0.21	->->-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	->->-> Unhiding rules:
0.00/0.21	 Empty
0.00/0.21	
0.00/0.21	
0.00/0.21	The problem is decomposed in 3 subproblems.
0.00/0.21	
0.00/0.21	Problem 1.1: 
0.00/0.21	
0.00/0.21	Reduction Pairs Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 IF(false,X,Y) -> Y
0.00/0.21	 IF(true,X,Y) -> X
0.00/0.21	 MINUS(X,Y) -> IF(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding rules:
0.00/0.21	 minus(p(X),p(Y)) -> MINUS(p(X),p(Y))
0.00/0.21	-> Usable rules:
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	->Interpretation type:
0.00/0.21	 Linear
0.00/0.21	->Coefficients:
0.00/0.21	 All rationals
0.00/0.21	->Dimension:
0.00/0.21	 1
0.00/0.21	->Bound:
0.00/0.21	 2
0.00/0.21	->Interpretation:
0.00/0.21	 
0.00/0.21	[gt](X1,X2) = X1
0.00/0.21	[minus](X1,X2) = 2.X1 + 2.X2 + 2
0.00/0.21	[p](X) = 1/2.X
0.00/0.21	[0] = 0
0.00/0.21	[false] = 0
0.00/0.21	[s](X) = 2.X + 1
0.00/0.21	[true] = 1/2
0.00/0.21	[IF](X1,X2,X3) = X1 + X2 + X3
0.00/0.21	[MINUS](X1,X2) = 2.X1 + 2.X2 + 2
0.00/0.21	
0.00/0.21	Problem 1.1: 
0.00/0.21	
0.00/0.21	SCC Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 IF(false,X,Y) -> Y
0.00/0.21	 MINUS(X,Y) -> IF(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding rules:
0.00/0.21	 minus(p(X),p(Y)) -> MINUS(p(X),p(Y))
0.00/0.21	->Strongly Connected Components:
0.00/0.21	->->Cycle:
0.00/0.21	->->-> Pairs:
0.00/0.21	 IF(false,X,Y) -> Y
0.00/0.21	 MINUS(X,Y) -> IF(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	->->-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	->->-> Unhiding rules:
0.00/0.21	 minus(p(X),p(Y)) -> MINUS(p(X),p(Y))
0.00/0.21	
0.00/0.21	Problem 1.1: 
0.00/0.21	
0.00/0.21	SubNColl Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 IF(false,X,Y) -> Y
0.00/0.21	 MINUS(X,Y) -> IF(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding rules:
0.00/0.21	 minus(p(X),p(Y)) -> MINUS(p(X),p(Y))
0.00/0.21	->Projection:
0.00/0.21	 pi(IF) = 3
0.00/0.21	 pi(MINUS) = 1
0.00/0.21	
0.00/0.21	Problem 1.1: 
0.00/0.21	
0.00/0.21	SCC Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 IF(false,X,Y) -> Y
0.00/0.21	 MINUS(X,Y) -> IF(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding rules:
0.00/0.21	 Empty
0.00/0.21	->Strongly Connected Components:
0.00/0.21	 There is no strongly connected component
0.00/0.21	
0.00/0.21	The problem is finite.
0.00/0.21	
0.00/0.21	Problem 1.2: 
0.00/0.21	
0.00/0.21	SubNColl Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 GT(s(X),s(Y)) -> GT(X,Y)
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding rules:
0.00/0.21	 Empty
0.00/0.21	->Projection:
0.00/0.21	 pi(GT) = 1
0.00/0.21	
0.00/0.21	Problem 1.2: 
0.00/0.21	
0.00/0.21	Basic Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 Empty
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding rules:
0.00/0.21	 Empty
0.00/0.21	-> Result:
0.00/0.21	 Set P is empty
0.00/0.21	
0.00/0.21	The problem is finite.
0.00/0.21	
0.00/0.21	Problem 1.3: 
0.00/0.21	
0.00/0.21	Reduction Pairs Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 DIV(s(X),s(Y)) -> DIV(minus(X,Y),s(Y))
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding rules:
0.00/0.21	 Empty
0.00/0.21	-> Usable rules:
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	->Interpretation type:
0.00/0.21	 Linear
0.00/0.21	->Coefficients:
0.00/0.21	 All rationals
0.00/0.21	->Dimension:
0.00/0.21	 1
0.00/0.21	->Bound:
0.00/0.21	 2
0.00/0.21	->Interpretation:
0.00/0.21	 
0.00/0.21	[gt](X1,X2) = 2.X1 + 1/2.X2
0.00/0.21	[if](X1,X2,X3) = X2 + X3
0.00/0.21	[minus](X1,X2) = 2.X1 + 1/2
0.00/0.21	[p](X) = 1/2.X
0.00/0.21	[0] = 0
0.00/0.21	[false] = 0
0.00/0.21	[s](X) = 2.X + 1
0.00/0.21	[true] = 0
0.00/0.21	[DIV](X1,X2) = X1
0.00/0.21	
0.00/0.21	Problem 1.3: 
0.00/0.21	
0.00/0.21	Basic Processor:
0.00/0.21	-> Pairs:
0.00/0.21	 Empty
0.00/0.21	-> Rules:
0.00/0.21	 div(0,s(Y)) -> 0
0.00/0.21	 div(s(X),s(Y)) -> s(div(minus(X,Y),s(Y)))
0.00/0.21	 gt(0,Y) -> false
0.00/0.21	 gt(s(X),0) -> true
0.00/0.21	 gt(s(X),s(Y)) -> gt(X,Y)
0.00/0.21	 if(false,X,Y) -> Y
0.00/0.21	 if(true,X,Y) -> X
0.00/0.21	 minus(X,Y) -> if(gt(Y,0),minus(p(X),p(Y)),X)
0.00/0.21	 p(0) -> 0
0.00/0.21	 p(s(X)) -> X
0.00/0.21	-> Unhiding rules:
0.00/0.21	 Empty
0.00/0.21	-> Result:
0.00/0.21	 Set P is empty
0.00/0.21	
0.00/0.21	The problem is finite.
0.00/0.21	EOF