0.00/0.34	YES
0.00/0.34	
0.00/0.34	Problem 1: 
0.00/0.34	
0.00/0.34	(VAR x y)
0.00/0.34	(THEORY 
0.00/0.34	(C gcd))
0.00/0.34	(RULES
0.00/0.34	gcd(0,y) -> y
0.00/0.34	gcd(s(x),0) -> s(x)
0.00/0.34	gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	le(0,y) -> true
0.00/0.34	le(s(x),0) -> false
0.00/0.34	le(s(x),s(y)) -> le(x,y)
0.00/0.34	minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	minus(x,0) -> x
0.00/0.34	)
0.00/0.34	
0.00/0.34	Problem 1: 
0.00/0.34	
0.00/0.34	Dependency Pairs Processor:
0.00/0.34	-> FAxioms:
0.00/0.34	 GCD(x2,x3) = GCD(x3,x2)
0.00/0.34	-> Pairs:
0.00/0.34	 GCD(s(x),s(y)) -> IF_GCD(le(y,x),s(x),s(y))
0.00/0.34	 GCD(s(x),s(y)) -> LE(y,x)
0.00/0.34	 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
0.00/0.34	 IF_GCD(false,s(x),s(y)) -> MINUS(y,x)
0.00/0.34	 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
0.00/0.34	 IF_GCD(true,s(x),s(y)) -> MINUS(x,y)
0.00/0.34	 LE(s(x),s(y)) -> LE(x,y)
0.00/0.34	 MINUS(s(x),s(y)) -> MINUS(x,y)
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	
0.00/0.34	Problem 1: 
0.00/0.34	
0.00/0.34	SCC Processor:
0.00/0.34	-> FAxioms:
0.00/0.34	 GCD(x2,x3) = GCD(x3,x2)
0.00/0.34	-> Pairs:
0.00/0.34	 GCD(s(x),s(y)) -> IF_GCD(le(y,x),s(x),s(y))
0.00/0.34	 GCD(s(x),s(y)) -> LE(y,x)
0.00/0.34	 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
0.00/0.34	 IF_GCD(false,s(x),s(y)) -> MINUS(y,x)
0.00/0.34	 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
0.00/0.34	 IF_GCD(true,s(x),s(y)) -> MINUS(x,y)
0.00/0.34	 LE(s(x),s(y)) -> LE(x,y)
0.00/0.34	 MINUS(s(x),s(y)) -> MINUS(x,y)
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	->Strongly Connected Components:
0.00/0.34	->->Cycle:
0.00/0.34	->->-> Pairs:
0.00/0.34	 MINUS(s(x),s(y)) -> MINUS(x,y)
0.00/0.34	-> FAxioms:
0.00/0.34	 gcd(x2,x3) -> gcd(x3,x2)
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	->->-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	->->Cycle:
0.00/0.34	->->-> Pairs:
0.00/0.34	 LE(s(x),s(y)) -> LE(x,y)
0.00/0.34	-> FAxioms:
0.00/0.34	 gcd(x2,x3) -> gcd(x3,x2)
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	->->-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	->->Cycle:
0.00/0.34	->->-> Pairs:
0.00/0.34	 GCD(s(x),s(y)) -> IF_GCD(le(y,x),s(x),s(y))
0.00/0.34	 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
0.00/0.34	 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
0.00/0.34	-> FAxioms:
0.00/0.34	 gcd(x2,x3) -> gcd(x3,x2)
0.00/0.34	 GCD(x2,x3) -> GCD(x3,x2)
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	->->-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	
0.00/0.34	
0.00/0.34	The problem is decomposed in 3 subproblems.
0.00/0.34	
0.00/0.34	Problem 1.1: 
0.00/0.34	
0.00/0.34	Subterm Processor:
0.00/0.34	-> FAxioms:
0.00/0.34	 Empty
0.00/0.34	-> Pairs:
0.00/0.34	 MINUS(s(x),s(y)) -> MINUS(x,y)
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	->Projection:
0.00/0.34	 pi(MINUS) = [1]
0.00/0.34	
0.00/0.34	Problem 1.1: 
0.00/0.34	
0.00/0.34	SCC Processor:
0.00/0.34	-> FAxioms:
0.00/0.34	 Empty
0.00/0.34	-> Pairs:
0.00/0.34	 Empty
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	->Strongly Connected Components:
0.00/0.34	 There is no strongly connected component
0.00/0.34	
0.00/0.34	The problem is finite.
0.00/0.34	
0.00/0.34	Problem 1.2: 
0.00/0.34	
0.00/0.34	Subterm Processor:
0.00/0.34	-> FAxioms:
0.00/0.34	 Empty
0.00/0.34	-> Pairs:
0.00/0.34	 LE(s(x),s(y)) -> LE(x,y)
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	->Projection:
0.00/0.34	 pi(LE) = [1]
0.00/0.34	
0.00/0.34	Problem 1.2: 
0.00/0.34	
0.00/0.34	SCC Processor:
0.00/0.34	-> FAxioms:
0.00/0.34	 Empty
0.00/0.34	-> Pairs:
0.00/0.34	 Empty
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	->Strongly Connected Components:
0.00/0.34	 There is no strongly connected component
0.00/0.34	
0.00/0.34	The problem is finite.
0.00/0.34	
0.00/0.34	Problem 1.3: 
0.00/0.34	
0.00/0.34	Reduction Pairs Processor:
0.00/0.34	-> FAxioms:
0.00/0.34	 GCD(x2,x3) = GCD(x3,x2)
0.00/0.34	-> Pairs:
0.00/0.34	 GCD(s(x),s(y)) -> IF_GCD(le(y,x),s(x),s(y))
0.00/0.34	 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
0.00/0.34	 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	-> Usable Equations:
0.00/0.34	 Empty
0.00/0.34	-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> Usable Rules:
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	->Interpretation type:
0.00/0.34	 Linear
0.00/0.34	->Coefficients:
0.00/0.34	 Natural Numbers
0.00/0.34	->Dimension:
0.00/0.34	 1
0.00/0.34	->Bound:
0.00/0.34	 2
0.00/0.34	->Interpretation:
0.00/0.34	 
0.00/0.34	[gcd](X1,X2) = 0
0.00/0.34	[if_gcd](X1,X2,X3) = 0
0.00/0.34	[le](X1,X2) = X1 + 2.X2
0.00/0.34	[minus](X1,X2) = 2.X1
0.00/0.34	[0] = 0
0.00/0.34	[false] = 1
0.00/0.34	[s](X) = 2.X + 2
0.00/0.34	[true] = 0
0.00/0.34	[GCD](X1,X2) = 2.X1 + 2.X2 + 2
0.00/0.34	[IF_GCD](X1,X2,X3) = 2.X2 + 2.X3 + 1
0.00/0.34	[LE](X1,X2) = 0
0.00/0.34	[MINUS](X1,X2) = 0
0.00/0.34	
0.00/0.34	Problem 1.3: 
0.00/0.34	
0.00/0.34	SCC Processor:
0.00/0.34	-> FAxioms:
0.00/0.34	 GCD(x2,x3) = GCD(x3,x2)
0.00/0.34	-> Pairs:
0.00/0.34	 IF_GCD(false,s(x),s(y)) -> GCD(minus(y,x),s(x))
0.00/0.34	 IF_GCD(true,s(x),s(y)) -> GCD(minus(x,y),s(y))
0.00/0.34	-> EAxioms:
0.00/0.34	 gcd(x2,x3) = gcd(x3,x2)
0.00/0.34	-> Rules:
0.00/0.34	 gcd(0,y) -> y
0.00/0.34	 gcd(s(x),0) -> s(x)
0.00/0.34	 gcd(s(x),s(y)) -> if_gcd(le(y,x),s(x),s(y))
0.00/0.34	 if_gcd(false,s(x),s(y)) -> gcd(minus(y,x),s(x))
0.00/0.34	 if_gcd(true,s(x),s(y)) -> gcd(minus(x,y),s(y))
0.00/0.34	 le(0,y) -> true
0.00/0.34	 le(s(x),0) -> false
0.00/0.34	 le(s(x),s(y)) -> le(x,y)
0.00/0.34	 minus(s(x),s(y)) -> minus(x,y)
0.00/0.34	 minus(x,0) -> x
0.00/0.34	-> SRules:
0.00/0.34	 Empty
0.00/0.34	->Strongly Connected Components:
0.00/0.34	 There is no strongly connected component
0.00/0.34	
0.00/0.34	The problem is finite.
0.00/0.34	EOF