0.00/0.05	YES
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	(VAR M N V1 V2 X)
0.00/0.05	(STRATEGY CONTEXTSENSITIVE
0.00/0.05	(U11 1)
0.00/0.05	(U21 1)
0.00/0.05	(and 1)
0.00/0.05	(isNat)
0.00/0.05	(plus 1 2)
0.00/0.05	(0)
0.00/0.05	(s 1)
0.00/0.05	(tt)
0.00/0.05	)
0.00/0.05	(RULES
0.00/0.05	U11(tt,N) -> N
0.00/0.05	U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	and(tt,X) -> X
0.00/0.05	isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	isNat(0) -> tt
0.00/0.05	isNat(s(V1)) -> isNat(V1)
0.00/0.05	plus(N,0) -> U11(isNat(N),N)
0.00/0.05	plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	)
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	Dependency Pairs Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 U11#(tt,N) -> N
0.00/0.05	 U21#(tt,M,N) -> PLUS(N,M)
0.00/0.05	 U21#(tt,M,N) -> M
0.00/0.05	 U21#(tt,M,N) -> N
0.00/0.05	 AND(tt,X) -> X
0.00/0.05	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
0.00/0.05	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
0.00/0.05	 ISNAT(s(V1)) -> ISNAT(V1)
0.00/0.05	 PLUS(N,0) -> U11#(isNat(N),N)
0.00/0.05	 PLUS(N,0) -> ISNAT(N)
0.00/0.05	 PLUS(N,s(M)) -> U21#(and(isNat(M),isNat(N)),M,N)
0.00/0.05	 PLUS(N,s(M)) -> AND(isNat(M),isNat(N))
0.00/0.05	 PLUS(N,s(M)) -> ISNAT(M)
0.00/0.05	-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	-> Unhiding Rules:
0.00/0.05	 isNat(N) -> ISNAT(N)
0.00/0.05	 isNat(V2) -> ISNAT(V2)
0.00/0.05	
0.00/0.05	Problem 1: 
0.00/0.05	
0.00/0.05	SCC Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 U11#(tt,N) -> N
0.00/0.05	 U21#(tt,M,N) -> PLUS(N,M)
0.00/0.05	 U21#(tt,M,N) -> M
0.00/0.05	 U21#(tt,M,N) -> N
0.00/0.05	 AND(tt,X) -> X
0.00/0.05	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
0.00/0.05	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
0.00/0.05	 ISNAT(s(V1)) -> ISNAT(V1)
0.00/0.05	 PLUS(N,0) -> U11#(isNat(N),N)
0.00/0.05	 PLUS(N,0) -> ISNAT(N)
0.00/0.05	 PLUS(N,s(M)) -> U21#(and(isNat(M),isNat(N)),M,N)
0.00/0.05	 PLUS(N,s(M)) -> AND(isNat(M),isNat(N))
0.00/0.05	 PLUS(N,s(M)) -> ISNAT(M)
0.00/0.05	-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	-> Unhiding rules:
0.00/0.05	 isNat(N) -> ISNAT(N)
0.00/0.05	 isNat(V2) -> ISNAT(V2)
0.00/0.05	->Strongly Connected Components:
0.00/0.05	->->Cycle:
0.00/0.05	->->-> Pairs:
0.00/0.05	 AND(tt,X) -> X
0.00/0.05	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
0.00/0.05	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
0.00/0.05	 ISNAT(s(V1)) -> ISNAT(V1)
0.00/0.05	->->-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	->->-> Unhiding rules:
0.00/0.05	 isNat(N) -> ISNAT(N)
0.00/0.05	 isNat(V2) -> ISNAT(V2)
0.00/0.05	->->Cycle:
0.00/0.05	->->-> Pairs:
0.00/0.05	 U21#(tt,M,N) -> PLUS(N,M)
0.00/0.05	 PLUS(N,s(M)) -> U21#(and(isNat(M),isNat(N)),M,N)
0.00/0.05	->->-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	->->-> Unhiding rules:
0.00/0.05	 Empty
0.00/0.05	
0.00/0.05	
0.00/0.05	The problem is decomposed in 2 subproblems.
0.00/0.05	
0.00/0.05	Problem 1.1: 
0.00/0.05	
0.00/0.05	Reduction Pairs Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 AND(tt,X) -> X
0.00/0.05	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
0.00/0.05	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
0.00/0.05	 ISNAT(s(V1)) -> ISNAT(V1)
0.00/0.05	-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	-> Unhiding rules:
0.00/0.05	 isNat(N) -> ISNAT(N)
0.00/0.05	 isNat(V2) -> ISNAT(V2)
0.00/0.05	-> Usable rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	->Interpretation type:
0.00/0.05	 Linear
0.00/0.05	->Coefficients:
0.00/0.05	 Natural Numbers
0.00/0.05	->Dimension:
0.00/0.05	 1
0.00/0.05	->Bound:
0.00/0.05	 2
0.00/0.05	->Interpretation:
0.00/0.05	 
0.00/0.05	[U11](X1,X2) = 2.X2 + 2
0.00/0.05	[U21](X1,X2,X3) = 2.X2 + 2.X3 + 2
0.00/0.05	[and](X1,X2) = 2.X1 + X2
0.00/0.05	[isNat](X) = 2.X + 1
0.00/0.05	[plus](X1,X2) = 2.X1 + 2.X2 + 2
0.00/0.05	[0] = 2
0.00/0.05	[s](X) = X
0.00/0.05	[tt] = 2
0.00/0.05	[AND](X1,X2) = 2.X1 + 2.X2 + 1
0.00/0.05	[ISNAT](X) = 2.X + 1
0.00/0.05	
0.00/0.05	Problem 1.1: 
0.00/0.05	
0.00/0.05	SCC Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
0.00/0.05	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
0.00/0.05	 ISNAT(s(V1)) -> ISNAT(V1)
0.00/0.05	-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	-> Unhiding rules:
0.00/0.05	 isNat(N) -> ISNAT(N)
0.00/0.05	 isNat(V2) -> ISNAT(V2)
0.00/0.05	->Strongly Connected Components:
0.00/0.05	->->Cycle:
0.00/0.05	->->-> Pairs:
0.00/0.05	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
0.00/0.05	 ISNAT(s(V1)) -> ISNAT(V1)
0.00/0.05	->->-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	->->-> Unhiding rules:
0.00/0.05	 Empty
0.00/0.05	
0.00/0.05	Problem 1.1: 
0.00/0.05	
0.00/0.05	SubNColl Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
0.00/0.05	 ISNAT(s(V1)) -> ISNAT(V1)
0.00/0.05	-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	-> Unhiding rules:
0.00/0.05	 Empty
0.00/0.05	->Projection:
0.00/0.05	 pi(ISNAT) = 1
0.00/0.05	
0.00/0.05	Problem 1.1: 
0.00/0.05	
0.00/0.05	Basic Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 Empty
0.00/0.05	-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	-> Unhiding rules:
0.00/0.05	 Empty
0.00/0.05	-> Result:
0.00/0.05	 Set P is empty
0.00/0.05	
0.00/0.05	The problem is finite.
0.00/0.05	
0.00/0.05	Problem 1.2: 
0.00/0.05	
0.00/0.05	SubNColl Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 U21#(tt,M,N) -> PLUS(N,M)
0.00/0.05	 PLUS(N,s(M)) -> U21#(and(isNat(M),isNat(N)),M,N)
0.00/0.05	-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	-> Unhiding rules:
0.00/0.05	 Empty
0.00/0.05	->Projection:
0.00/0.05	 pi(U21#) = 2
0.00/0.05	 pi(PLUS) = 2
0.00/0.05	
0.00/0.05	Problem 1.2: 
0.00/0.05	
0.00/0.05	SCC Processor:
0.00/0.05	-> Pairs:
0.00/0.05	 U21#(tt,M,N) -> PLUS(N,M)
0.00/0.05	-> Rules:
0.00/0.05	 U11(tt,N) -> N
0.00/0.05	 U21(tt,M,N) -> s(plus(N,M))
0.00/0.05	 and(tt,X) -> X
0.00/0.05	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
0.00/0.05	 isNat(0) -> tt
0.00/0.05	 isNat(s(V1)) -> isNat(V1)
0.00/0.05	 plus(N,0) -> U11(isNat(N),N)
0.00/0.05	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
0.00/0.05	-> Unhiding rules:
0.00/0.05	 Empty
0.00/0.05	->Strongly Connected Components:
0.00/0.05	 There is no strongly connected component
0.00/0.05	
0.00/0.05	The problem is finite.
0.00/0.05	EOF