0.00/0.25	YES
0.00/0.25	
0.00/0.25	Problem 1: 
0.00/0.25	
0.00/0.25	(VAR M N V1 V2 X)
0.00/0.25	(STRATEGY CONTEXTSENSITIVE
0.00/0.25	(U11 1)
0.00/0.25	(U12 1)
0.00/0.25	(U13 1)
0.00/0.25	(U21 1)
0.00/0.25	(U22 1)
0.00/0.25	(U31 1)
0.00/0.25	(U41 1)
0.00/0.25	(and 1)
0.00/0.25	(isNat)
0.00/0.25	(isNatKind)
0.00/0.25	(plus 1 2)
0.00/0.25	(0)
0.00/0.25	(s 1)
0.00/0.25	(tt)
0.00/0.25	)
0.00/0.25	(RULES
0.00/0.25	U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	U13(tt) -> tt
0.00/0.25	U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	U22(tt) -> tt
0.00/0.25	U31(tt,N) -> N
0.00/0.25	U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	and(tt,X) -> X
0.00/0.25	isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	isNat(0) -> tt
0.00/0.25	isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	isNatKind(0) -> tt
0.00/0.25	isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	)
0.00/0.25	
0.00/0.25	Problem 1: 
0.00/0.25	
0.00/0.25	Dependency Pairs Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
0.00/0.25	 U11#(tt,V1,V2) -> ISNAT(V1)
0.00/0.25	 U12#(tt,V2) -> U13#(isNat(V2))
0.00/0.25	 U12#(tt,V2) -> ISNAT(V2)
0.00/0.25	 U21#(tt,V1) -> U22#(isNat(V1))
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 U31#(tt,N) -> N
0.00/0.25	 U41#(tt,M,N) -> PLUS(N,M)
0.00/0.25	 U41#(tt,M,N) -> M
0.00/0.25	 U41#(tt,M,N) -> N
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 PLUS(N,0) -> U31#(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 PLUS(N,0) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 PLUS(N,0) -> ISNAT(N)
0.00/0.25	 PLUS(N,s(M)) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	 PLUS(N,s(M)) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
0.00/0.25	 PLUS(N,s(M)) -> AND(isNat(M),isNatKind(M))
0.00/0.25	 PLUS(N,s(M)) -> ISNAT(M)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding Rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	
0.00/0.25	Problem 1: 
0.00/0.25	
0.00/0.25	SCC Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
0.00/0.25	 U11#(tt,V1,V2) -> ISNAT(V1)
0.00/0.25	 U12#(tt,V2) -> U13#(isNat(V2))
0.00/0.25	 U12#(tt,V2) -> ISNAT(V2)
0.00/0.25	 U21#(tt,V1) -> U22#(isNat(V1))
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 U31#(tt,N) -> N
0.00/0.25	 U41#(tt,M,N) -> PLUS(N,M)
0.00/0.25	 U41#(tt,M,N) -> M
0.00/0.25	 U41#(tt,M,N) -> N
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 PLUS(N,0) -> U31#(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 PLUS(N,0) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 PLUS(N,0) -> ISNAT(N)
0.00/0.25	 PLUS(N,s(M)) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	 PLUS(N,s(M)) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
0.00/0.25	 PLUS(N,s(M)) -> AND(isNat(M),isNatKind(M))
0.00/0.25	 PLUS(N,s(M)) -> ISNAT(M)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	->Strongly Connected Components:
0.00/0.25	->->Cycle:
0.00/0.25	->->-> Pairs:
0.00/0.25	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
0.00/0.25	 U11#(tt,V1,V2) -> ISNAT(V1)
0.00/0.25	 U12#(tt,V2) -> ISNAT(V2)
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	->->-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->->-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	->->Cycle:
0.00/0.25	->->-> Pairs:
0.00/0.25	 U41#(tt,M,N) -> PLUS(N,M)
0.00/0.25	 PLUS(N,s(M)) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->->-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->->-> Unhiding rules:
0.00/0.25	 Empty
0.00/0.25	
0.00/0.25	
0.00/0.25	The problem is decomposed in 2 subproblems.
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	Reduction Pairs Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
0.00/0.25	 U11#(tt,V1,V2) -> ISNAT(V1)
0.00/0.25	 U12#(tt,V2) -> ISNAT(V2)
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	-> Usable rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->Interpretation type:
0.00/0.25	 Linear
0.00/0.25	->Coefficients:
0.00/0.25	 Natural Numbers
0.00/0.25	->Dimension:
0.00/0.25	 1
0.00/0.25	->Bound:
0.00/0.25	 2
0.00/0.25	->Interpretation:
0.00/0.25	 
0.00/0.25	[U11](X1,X2,X3) = 2
0.00/0.25	[U12](X1,X2) = X1
0.00/0.25	[U13](X) = 2
0.00/0.25	[U21](X1,X2) = 2
0.00/0.25	[U22](X) = 2
0.00/0.25	[U31](X1,X2) = 2.X2
0.00/0.25	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
0.00/0.25	[and](X1,X2) = X2 + 2
0.00/0.25	[isNat](X) = 2
0.00/0.25	[isNatKind](X) = 2.X + 2
0.00/0.25	[plus](X1,X2) = 2.X1 + 2.X2 + 1
0.00/0.25	[0] = 2
0.00/0.25	[s](X) = X + 1
0.00/0.25	[tt] = 2
0.00/0.25	[U11#](X1,X2,X3) = 2.X2 + 2.X3 + 2
0.00/0.25	[U12#](X1,X2) = 2.X2 + 1
0.00/0.25	[U21#](X1,X2) = 2.X2
0.00/0.25	[AND](X1,X2) = X2
0.00/0.25	[ISNAT](X) = 2.X
0.00/0.25	[ISNATKIND](X) = 2.X + 2
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	SCC Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 U11#(tt,V1,V2) -> ISNAT(V1)
0.00/0.25	 U12#(tt,V2) -> ISNAT(V2)
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	->Strongly Connected Components:
0.00/0.25	->->Cycle:
0.00/0.25	->->-> Pairs:
0.00/0.25	 U11#(tt,V1,V2) -> ISNAT(V1)
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	->->-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->->-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	Reduction Pairs Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 U11#(tt,V1,V2) -> ISNAT(V1)
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	-> Usable rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->Interpretation type:
0.00/0.25	 Linear
0.00/0.25	->Coefficients:
0.00/0.25	 Natural Numbers
0.00/0.25	->Dimension:
0.00/0.25	 1
0.00/0.25	->Bound:
0.00/0.25	 2
0.00/0.25	->Interpretation:
0.00/0.25	 
0.00/0.25	[U11](X1,X2,X3) = X1 + 2.X2 + 2
0.00/0.25	[U12](X1,X2) = X1 + 2
0.00/0.25	[U13](X) = 2
0.00/0.25	[U21](X1,X2) = 2.X2 + 2
0.00/0.25	[U22](X) = X
0.00/0.25	[U31](X1,X2) = X2
0.00/0.25	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
0.00/0.25	[and](X1,X2) = X1 + 2.X2 + 1
0.00/0.25	[isNat](X) = 2.X + 2
0.00/0.25	[isNatKind](X) = 2.X + 1
0.00/0.25	[plus](X1,X2) = 2.X1 + 2.X2 + 2
0.00/0.25	[0] = 2
0.00/0.25	[s](X) = X
0.00/0.25	[tt] = 2
0.00/0.25	[U11#](X1,X2,X3) = X1 + 2.X2
0.00/0.25	[U21#](X1,X2) = 2.X2 + 1
0.00/0.25	[AND](X1,X2) = 2.X1 + X2
0.00/0.25	[ISNAT](X) = 2.X + 1
0.00/0.25	[ISNATKIND](X) = 2.X + 1
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	SCC Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	->Strongly Connected Components:
0.00/0.25	->->Cycle:
0.00/0.25	->->-> Pairs:
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	->->-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->->-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	Reduction Pairs Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 U21#(tt,V1) -> ISNAT(V1)
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	-> Usable rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->Interpretation type:
0.00/0.25	 Linear
0.00/0.25	->Coefficients:
0.00/0.25	 Natural Numbers
0.00/0.25	->Dimension:
0.00/0.25	 1
0.00/0.25	->Bound:
0.00/0.25	 2
0.00/0.25	->Interpretation:
0.00/0.25	 
0.00/0.25	[U11](X1,X2,X3) = 2.X2 + 2.X3 + 2
0.00/0.25	[U12](X1,X2) = 2
0.00/0.25	[U13](X) = 2
0.00/0.25	[U21](X1,X2) = X1 + 2.X2 + 2
0.00/0.25	[U22](X) = X + 1
0.00/0.25	[U31](X1,X2) = X2 + 1
0.00/0.25	[U41](X1,X2,X3) = 2.X2 + X3 + 2
0.00/0.25	[and](X1,X2) = 2.X1 + 2.X2
0.00/0.25	[isNat](X) = 2.X + 1
0.00/0.25	[isNatKind](X) = 0
0.00/0.25	[plus](X1,X2) = X1 + 2.X2 + 1
0.00/0.25	[0] = 0
0.00/0.25	[s](X) = X + 1
0.00/0.25	[tt] = 0
0.00/0.25	[U21#](X1,X2) = 2.X2 + 2
0.00/0.25	[AND](X1,X2) = X2
0.00/0.25	[ISNAT](X) = 2.X + 1
0.00/0.25	[ISNATKIND](X) = 0
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	SCC Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	->Strongly Connected Components:
0.00/0.25	->->Cycle:
0.00/0.25	->->-> Pairs:
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	->->-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->->-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	Reduction Pairs Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 AND(tt,X) -> X
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	-> Usable rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->Interpretation type:
0.00/0.25	 Linear
0.00/0.25	->Coefficients:
0.00/0.25	 Natural Numbers
0.00/0.25	->Dimension:
0.00/0.25	 1
0.00/0.25	->Bound:
0.00/0.25	 2
0.00/0.25	->Interpretation:
0.00/0.25	 
0.00/0.25	[U11](X1,X2,X3) = X1 + 2.X2
0.00/0.25	[U12](X1,X2) = X1
0.00/0.25	[U13](X) = 2
0.00/0.25	[U21](X1,X2) = 2
0.00/0.25	[U22](X) = 2
0.00/0.25	[U31](X1,X2) = 2.X2
0.00/0.25	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
0.00/0.25	[and](X1,X2) = X2 + 1
0.00/0.25	[isNat](X) = X + 2
0.00/0.25	[isNatKind](X) = 2.X + 2
0.00/0.25	[plus](X1,X2) = 2.X1 + 2.X2 + 1
0.00/0.25	[0] = 1
0.00/0.25	[s](X) = X + 1
0.00/0.25	[tt] = 2
0.00/0.25	[AND](X1,X2) = X2 + 1
0.00/0.25	[ISNAT](X) = 2.X + 1
0.00/0.25	[ISNATKIND](X) = 2.X + 2
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	SCC Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNAT(s(V1)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
0.00/0.25	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
0.00/0.25	 isNatKind(M) -> ISNATKIND(M)
0.00/0.25	 isNatKind(N) -> ISNATKIND(N)
0.00/0.25	 isNatKind(V2) -> ISNATKIND(V2)
0.00/0.25	->Strongly Connected Components:
0.00/0.25	->->Cycle:
0.00/0.25	->->-> Pairs:
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	->->-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	->->-> Unhiding rules:
0.00/0.25	 Empty
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	SubNColl Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
0.00/0.25	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 Empty
0.00/0.25	->Projection:
0.00/0.25	 pi(ISNATKIND) = 1
0.00/0.25	
0.00/0.25	Problem 1.1: 
0.00/0.25	
0.00/0.25	Basic Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 Empty
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 Empty
0.00/0.25	-> Result:
0.00/0.25	 Set P is empty
0.00/0.25	
0.00/0.25	The problem is finite.
0.00/0.25	
0.00/0.25	Problem 1.2: 
0.00/0.25	
0.00/0.25	SubNColl Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 U41#(tt,M,N) -> PLUS(N,M)
0.00/0.25	 PLUS(N,s(M)) -> U41#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 Empty
0.00/0.25	->Projection:
0.00/0.25	 pi(U41#) = 2
0.00/0.25	 pi(PLUS) = 2
0.00/0.25	
0.00/0.25	Problem 1.2: 
0.00/0.25	
0.00/0.25	SCC Processor:
0.00/0.25	-> Pairs:
0.00/0.25	 U41#(tt,M,N) -> PLUS(N,M)
0.00/0.25	-> Rules:
0.00/0.25	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
0.00/0.25	 U12(tt,V2) -> U13(isNat(V2))
0.00/0.25	 U13(tt) -> tt
0.00/0.25	 U21(tt,V1) -> U22(isNat(V1))
0.00/0.25	 U22(tt) -> tt
0.00/0.25	 U31(tt,N) -> N
0.00/0.25	 U41(tt,M,N) -> s(plus(N,M))
0.00/0.25	 and(tt,X) -> X
0.00/0.25	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
0.00/0.25	 isNat(0) -> tt
0.00/0.25	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
0.00/0.25	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
0.00/0.25	 isNatKind(0) -> tt
0.00/0.25	 isNatKind(s(V1)) -> isNatKind(V1)
0.00/0.25	 plus(N,0) -> U31(and(isNat(N),isNatKind(N)),N)
0.00/0.25	 plus(N,s(M)) -> U41(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
0.00/0.25	-> Unhiding rules:
0.00/0.25	 Empty
0.00/0.25	->Strongly Connected Components:
0.00/0.25	 There is no strongly connected component
0.00/0.25	
0.00/0.25	The problem is finite.
0.00/0.25	EOF