1.36/1.58	YES
1.36/1.58	
1.36/1.58	Problem 1: 
1.36/1.58	
1.36/1.58	(VAR M N V1 V2 X)
1.36/1.58	(STRATEGY CONTEXTSENSITIVE
1.36/1.58	(U11 1)
1.36/1.58	(U21 1)
1.36/1.58	(U31 1)
1.36/1.58	(U41 1)
1.36/1.58	(and 1)
1.36/1.58	(isNat)
1.36/1.58	(plus 1 2)
1.36/1.58	(x 1 2)
1.36/1.58	(0)
1.36/1.58	(s 1)
1.36/1.58	(tt)
1.36/1.58	)
1.36/1.58	(RULES
1.36/1.58	U11(tt,N) -> N
1.36/1.58	U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	U31(tt) -> 0
1.36/1.58	U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	and(tt,X) -> X
1.36/1.58	isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	isNat(0) -> tt
1.36/1.58	isNat(s(V1)) -> isNat(V1)
1.36/1.58	plus(N,0) -> U11(isNat(N),N)
1.36/1.58	plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	x(N,0) -> U31(isNat(N))
1.36/1.58	x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	)
1.36/1.58	
1.36/1.58	Problem 1: 
1.36/1.58	
1.36/1.58	Dependency Pairs Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 U11#(tt,N) -> N
1.36/1.58	 U21#(tt,M,N) -> PLUS(N,M)
1.36/1.58	 U21#(tt,M,N) -> M
1.36/1.58	 U21#(tt,M,N) -> N
1.36/1.58	 U41#(tt,M,N) -> PLUS(x(N,M),N)
1.36/1.58	 U41#(tt,M,N) -> X(N,M)
1.36/1.58	 U41#(tt,M,N) -> M
1.36/1.58	 U41#(tt,M,N) -> N
1.36/1.58	 AND(tt,X) -> X
1.36/1.58	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(x(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(x(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(s(V1)) -> ISNAT(V1)
1.36/1.58	 PLUS(N,0) -> U11#(isNat(N),N)
1.36/1.58	 PLUS(N,0) -> ISNAT(N)
1.36/1.58	 PLUS(N,s(M)) -> U21#(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 PLUS(N,s(M)) -> AND(isNat(M),isNat(N))
1.36/1.58	 PLUS(N,s(M)) -> ISNAT(M)
1.36/1.58	 X(N,0) -> U31#(isNat(N))
1.36/1.58	 X(N,0) -> ISNAT(N)
1.36/1.58	 X(N,s(M)) -> U41#(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 X(N,s(M)) -> AND(isNat(M),isNat(N))
1.36/1.58	 X(N,s(M)) -> ISNAT(M)
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding Rules:
1.36/1.58	 isNat(N) -> ISNAT(N)
1.36/1.58	 isNat(V2) -> ISNAT(V2)
1.36/1.58	
1.36/1.58	Problem 1: 
1.36/1.58	
1.36/1.58	SCC Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 U11#(tt,N) -> N
1.36/1.58	 U21#(tt,M,N) -> PLUS(N,M)
1.36/1.58	 U21#(tt,M,N) -> M
1.36/1.58	 U21#(tt,M,N) -> N
1.36/1.58	 U41#(tt,M,N) -> PLUS(x(N,M),N)
1.36/1.58	 U41#(tt,M,N) -> X(N,M)
1.36/1.58	 U41#(tt,M,N) -> M
1.36/1.58	 U41#(tt,M,N) -> N
1.36/1.58	 AND(tt,X) -> X
1.36/1.58	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(x(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(x(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(s(V1)) -> ISNAT(V1)
1.36/1.58	 PLUS(N,0) -> U11#(isNat(N),N)
1.36/1.58	 PLUS(N,0) -> ISNAT(N)
1.36/1.58	 PLUS(N,s(M)) -> U21#(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 PLUS(N,s(M)) -> AND(isNat(M),isNat(N))
1.36/1.58	 PLUS(N,s(M)) -> ISNAT(M)
1.36/1.58	 X(N,0) -> U31#(isNat(N))
1.36/1.58	 X(N,0) -> ISNAT(N)
1.36/1.58	 X(N,s(M)) -> U41#(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 X(N,s(M)) -> AND(isNat(M),isNat(N))
1.36/1.58	 X(N,s(M)) -> ISNAT(M)
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding rules:
1.36/1.58	 isNat(N) -> ISNAT(N)
1.36/1.58	 isNat(V2) -> ISNAT(V2)
1.36/1.58	->Strongly Connected Components:
1.36/1.58	->->Cycle:
1.36/1.58	->->-> Pairs:
1.36/1.58	 AND(tt,X) -> X
1.36/1.58	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(x(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(x(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(s(V1)) -> ISNAT(V1)
1.36/1.58	->->-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	->->-> Unhiding rules:
1.36/1.58	 isNat(N) -> ISNAT(N)
1.36/1.58	 isNat(V2) -> ISNAT(V2)
1.36/1.58	->->Cycle:
1.36/1.58	->->-> Pairs:
1.36/1.58	 U21#(tt,M,N) -> PLUS(N,M)
1.36/1.58	 PLUS(N,s(M)) -> U21#(and(isNat(M),isNat(N)),M,N)
1.36/1.58	->->-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	->->-> Unhiding rules:
1.36/1.58	 Empty
1.36/1.58	->->Cycle:
1.36/1.58	->->-> Pairs:
1.36/1.58	 U41#(tt,M,N) -> X(N,M)
1.36/1.58	 X(N,s(M)) -> U41#(and(isNat(M),isNat(N)),M,N)
1.36/1.58	->->-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	->->-> Unhiding rules:
1.36/1.58	 Empty
1.36/1.58	
1.36/1.58	
1.36/1.58	The problem is decomposed in 3 subproblems.
1.36/1.58	
1.36/1.58	Problem 1.1: 
1.36/1.58	
1.36/1.58	Reduction Pairs Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 AND(tt,X) -> X
1.36/1.58	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(x(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(x(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(s(V1)) -> ISNAT(V1)
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding rules:
1.36/1.58	 isNat(N) -> ISNAT(N)
1.36/1.58	 isNat(V2) -> ISNAT(V2)
1.36/1.58	-> Usable rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	->Interpretation type:
1.36/1.58	 Simple mixed
1.36/1.58	->Coefficients:
1.36/1.58	 Natural Numbers
1.36/1.58	->Dimension:
1.36/1.58	 1
1.36/1.58	->Bound:
1.36/1.58	 2
1.36/1.58	->Interpretation:
1.36/1.58	 
1.36/1.58	[U11](X1,X2) = X2 + 2
1.36/1.58	[U21](X1,X2,X3) = X2 + X3 + 2
1.36/1.58	[U31](X) = X + 2
1.36/1.58	[U41](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2
1.36/1.58	[and](X1,X2) = X2
1.36/1.58	[isNat](X) = X + 2
1.36/1.58	[plus](X1,X2) = X1 + X2 + 1
1.36/1.58	[x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 2
1.36/1.58	[0] = 2
1.36/1.58	[s](X) = X + 1
1.36/1.58	[tt] = 2
1.36/1.58	[AND](X1,X2) = X2 + 1
1.36/1.58	[ISNAT](X) = X + 2
1.36/1.58	
1.36/1.58	Problem 1.1: 
1.36/1.58	
1.36/1.58	SCC Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 ISNAT(plus(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(x(V1,V2)) -> AND(isNat(V1),isNat(V2))
1.36/1.58	 ISNAT(x(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(s(V1)) -> ISNAT(V1)
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding rules:
1.36/1.58	 isNat(N) -> ISNAT(N)
1.36/1.58	 isNat(V2) -> ISNAT(V2)
1.36/1.58	->Strongly Connected Components:
1.36/1.58	->->Cycle:
1.36/1.58	->->-> Pairs:
1.36/1.58	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(x(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(s(V1)) -> ISNAT(V1)
1.36/1.58	->->-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	->->-> Unhiding rules:
1.36/1.58	 Empty
1.36/1.58	
1.36/1.58	Problem 1.1: 
1.36/1.58	
1.36/1.58	SubNColl Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 ISNAT(plus(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(x(V1,V2)) -> ISNAT(V1)
1.36/1.58	 ISNAT(s(V1)) -> ISNAT(V1)
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding rules:
1.36/1.58	 Empty
1.36/1.58	->Projection:
1.36/1.58	 pi(ISNAT) = 1
1.36/1.58	
1.36/1.58	Problem 1.1: 
1.36/1.58	
1.36/1.58	Basic Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 Empty
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding rules:
1.36/1.58	 Empty
1.36/1.58	-> Result:
1.36/1.58	 Set P is empty
1.36/1.58	
1.36/1.58	The problem is finite.
1.36/1.58	
1.36/1.58	Problem 1.2: 
1.36/1.58	
1.36/1.58	SubNColl Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 U21#(tt,M,N) -> PLUS(N,M)
1.36/1.58	 PLUS(N,s(M)) -> U21#(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding rules:
1.36/1.58	 Empty
1.36/1.58	->Projection:
1.36/1.58	 pi(U21#) = 2
1.36/1.58	 pi(PLUS) = 2
1.36/1.58	
1.36/1.58	Problem 1.2: 
1.36/1.58	
1.36/1.58	SCC Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 U21#(tt,M,N) -> PLUS(N,M)
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding rules:
1.36/1.58	 Empty
1.36/1.58	->Strongly Connected Components:
1.36/1.58	 There is no strongly connected component
1.36/1.58	
1.36/1.58	The problem is finite.
1.36/1.58	
1.36/1.58	Problem 1.3: 
1.36/1.58	
1.36/1.58	SubNColl Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 U41#(tt,M,N) -> X(N,M)
1.36/1.58	 X(N,s(M)) -> U41#(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding rules:
1.36/1.58	 Empty
1.36/1.58	->Projection:
1.36/1.58	 pi(U41#) = 2
1.36/1.58	 pi(X) = 2
1.36/1.58	
1.36/1.58	Problem 1.3: 
1.36/1.58	
1.36/1.58	SCC Processor:
1.36/1.58	-> Pairs:
1.36/1.58	 U41#(tt,M,N) -> X(N,M)
1.36/1.58	-> Rules:
1.36/1.58	 U11(tt,N) -> N
1.36/1.58	 U21(tt,M,N) -> s(plus(N,M))
1.36/1.58	 U31(tt) -> 0
1.36/1.58	 U41(tt,M,N) -> plus(x(N,M),N)
1.36/1.58	 and(tt,X) -> X
1.36/1.58	 isNat(plus(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(x(V1,V2)) -> and(isNat(V1),isNat(V2))
1.36/1.58	 isNat(0) -> tt
1.36/1.58	 isNat(s(V1)) -> isNat(V1)
1.36/1.58	 plus(N,0) -> U11(isNat(N),N)
1.36/1.58	 plus(N,s(M)) -> U21(and(isNat(M),isNat(N)),M,N)
1.36/1.58	 x(N,0) -> U31(isNat(N))
1.36/1.58	 x(N,s(M)) -> U41(and(isNat(M),isNat(N)),M,N)
1.36/1.58	-> Unhiding rules:
1.36/1.58	 Empty
1.36/1.58	->Strongly Connected Components:
1.36/1.58	 There is no strongly connected component
1.36/1.58	
1.36/1.58	The problem is finite.
1.36/1.58	EOF