1.06/1.35	YES
1.06/1.35	
1.06/1.35	Problem 1: 
1.06/1.35	
1.06/1.35	(VAR x y)
1.06/1.35	(STRATEGY CONTEXTSENSITIVE
1.06/1.35	(if 1)
1.06/1.35	(isZero 1)
1.06/1.35	(p 1)
1.06/1.35	(plus 1 2)
1.06/1.35	(0)
1.06/1.35	(false)
1.06/1.35	(s 1)
1.06/1.35	(true)
1.06/1.35	)
1.06/1.35	(RULES
1.06/1.35	if(false,x,y) -> y
1.06/1.35	if(true,x,y) -> x
1.06/1.35	isZero(0) -> true
1.06/1.35	isZero(s(x)) -> false
1.06/1.35	p(s(x)) -> x
1.06/1.35	plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	)
1.06/1.35	
1.06/1.35	Problem 1: 
1.06/1.35	
1.06/1.35	Innermost Equivalent Processor:
1.06/1.35	-> Rules:
1.06/1.35	 if(false,x,y) -> y
1.06/1.35	 if(true,x,y) -> x
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	 plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> The context-sensitive term rewriting system is an orthogonal system. Therefore, innermost cs-termination implies cs-termination.
1.06/1.35	 
1.06/1.35	
1.06/1.35	Problem 1: 
1.06/1.35	
1.06/1.35	Dependency Pairs Processor:
1.06/1.35	-> Pairs:
1.06/1.35	 IF(false,x,y) -> y
1.06/1.35	 IF(true,x,y) -> x
1.06/1.35	 PLUS(x,y) -> IF(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	 PLUS(x,y) -> ISZERO(x)
1.06/1.35	-> Rules:
1.06/1.35	 if(false,x,y) -> y
1.06/1.35	 if(true,x,y) -> x
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	 plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Unhiding Rules:
1.06/1.35	 s(plus(p(x),y)) -> P(x)
1.06/1.35	 s(plus(p(x),y)) -> PLUS(p(x),y)
1.06/1.35	
1.06/1.35	Problem 1: 
1.06/1.35	
1.06/1.35	SCC Processor:
1.06/1.35	-> Pairs:
1.06/1.35	 IF(false,x,y) -> y
1.06/1.35	 IF(true,x,y) -> x
1.06/1.35	 PLUS(x,y) -> IF(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	 PLUS(x,y) -> ISZERO(x)
1.06/1.35	-> Rules:
1.06/1.35	 if(false,x,y) -> y
1.06/1.35	 if(true,x,y) -> x
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	 plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Unhiding rules:
1.06/1.35	 s(plus(p(x),y)) -> P(x)
1.06/1.35	 s(plus(p(x),y)) -> PLUS(p(x),y)
1.06/1.35	->Strongly Connected Components:
1.06/1.35	->->Cycle:
1.06/1.35	->->-> Pairs:
1.06/1.35	 IF(false,x,y) -> y
1.06/1.35	 IF(true,x,y) -> x
1.06/1.35	 PLUS(x,y) -> IF(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	->->-> Rules:
1.06/1.35	 if(false,x,y) -> y
1.06/1.35	 if(true,x,y) -> x
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	 plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	->->-> Unhiding rules:
1.06/1.35	 s(plus(p(x),y)) -> PLUS(p(x),y)
1.06/1.35	
1.06/1.35	Problem 1: 
1.06/1.35	
1.06/1.35	Reduction Pairs Processor:
1.06/1.35	-> Pairs:
1.06/1.35	 IF(false,x,y) -> y
1.06/1.35	 IF(true,x,y) -> x
1.06/1.35	 PLUS(x,y) -> IF(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Rules:
1.06/1.35	 if(false,x,y) -> y
1.06/1.35	 if(true,x,y) -> x
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	 plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Unhiding rules:
1.06/1.35	 s(plus(p(x),y)) -> PLUS(p(x),y)
1.06/1.35	-> Usable rules:
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	->Interpretation type:
1.06/1.35	 Simple mixed
1.06/1.35	->Coefficients:
1.06/1.35	 All rationals
1.06/1.35	->Dimension:
1.06/1.35	 1
1.06/1.35	->Bound:
1.06/1.35	 2
1.06/1.35	->Interpretation:
1.06/1.35	 
1.06/1.35	[isZero](X) = 2.X
1.06/1.35	[p](X) = 1/2.X
1.06/1.35	[plus](X1,X2) = X1.X2 + X1 + 1/2
1.06/1.35	[0] = 2
1.06/1.35	[false] = 2
1.06/1.35	[s](X) = 2.X + 1
1.06/1.35	[true] = 2
1.06/1.35	[IF](X1,X2,X3) = 1/2.X1.X2 + 1/2.X1 + X3
1.06/1.35	[PLUS](X1,X2) = 2.X1.X2 + 2.X1 + 2
1.06/1.35	
1.06/1.35	Problem 1: 
1.06/1.35	
1.06/1.35	SCC Processor:
1.06/1.35	-> Pairs:
1.06/1.35	 IF(true,x,y) -> x
1.06/1.35	 PLUS(x,y) -> IF(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Rules:
1.06/1.35	 if(false,x,y) -> y
1.06/1.35	 if(true,x,y) -> x
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	 plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Unhiding rules:
1.06/1.35	 s(plus(p(x),y)) -> PLUS(p(x),y)
1.06/1.35	->Strongly Connected Components:
1.06/1.35	->->Cycle:
1.06/1.35	->->-> Pairs:
1.06/1.35	 IF(true,x,y) -> x
1.06/1.35	 PLUS(x,y) -> IF(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	->->-> Rules:
1.06/1.35	 if(false,x,y) -> y
1.06/1.35	 if(true,x,y) -> x
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	 plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	->->-> Unhiding rules:
1.06/1.35	 s(plus(p(x),y)) -> PLUS(p(x),y)
1.06/1.35	
1.06/1.35	Problem 1: 
1.06/1.35	
1.06/1.35	SubNColl Processor:
1.06/1.35	-> Pairs:
1.06/1.35	 IF(true,x,y) -> x
1.06/1.35	 PLUS(x,y) -> IF(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Rules:
1.06/1.35	 if(false,x,y) -> y
1.06/1.35	 if(true,x,y) -> x
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	 plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Unhiding rules:
1.06/1.35	 s(plus(p(x),y)) -> PLUS(p(x),y)
1.06/1.35	->Projection:
1.06/1.35	 pi(IF) = 2
1.06/1.35	 pi(PLUS) = 2
1.06/1.35	
1.06/1.35	Problem 1: 
1.06/1.35	
1.06/1.35	SCC Processor:
1.06/1.35	-> Pairs:
1.06/1.35	 IF(true,x,y) -> x
1.06/1.35	 PLUS(x,y) -> IF(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Rules:
1.06/1.35	 if(false,x,y) -> y
1.06/1.35	 if(true,x,y) -> x
1.06/1.35	 isZero(0) -> true
1.06/1.35	 isZero(s(x)) -> false
1.06/1.35	 p(s(x)) -> x
1.06/1.35	 plus(x,y) -> if(isZero(x),y,s(plus(p(x),y)))
1.06/1.35	-> Unhiding rules:
1.06/1.35	 Empty
1.06/1.35	->Strongly Connected Components:
1.06/1.35	 There is no strongly connected component
1.06/1.35	
1.06/1.35	The problem is finite.
1.29/1.38	EOF