14.03/15.00	YES
14.03/15.00	
14.03/15.00	Problem 1: 
14.03/15.00	
14.03/15.00	(VAR M N V1 V2 X)
14.03/15.00	(STRATEGY CONTEXTSENSITIVE
14.03/15.00	(U11 1)
14.03/15.00	(U12 1)
14.03/15.00	(U13 1)
14.03/15.00	(U21 1)
14.03/15.00	(U22 1)
14.03/15.00	(U31 1)
14.03/15.00	(U32 1)
14.03/15.00	(U33 1)
14.03/15.00	(U41 1)
14.03/15.00	(U51 1)
14.03/15.00	(U61 1)
14.03/15.00	(U71 1)
14.03/15.00	(and 1)
14.03/15.00	(isNat)
14.03/15.00	(isNatKind)
14.03/15.00	(plus 1 2)
14.03/15.00	(x 1 2)
14.03/15.00	(0)
14.03/15.00	(s 1)
14.03/15.00	(tt)
14.03/15.00	)
14.03/15.00	(RULES
14.03/15.00	U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	U13(tt) -> tt
14.03/15.00	U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	U22(tt) -> tt
14.03/15.00	U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	U33(tt) -> tt
14.03/15.00	U41(tt,N) -> N
14.03/15.00	U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	U61(tt) -> 0
14.03/15.00	U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	and(tt,X) -> X
14.03/15.00	isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	isNat(0) -> tt
14.03/15.00	isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	isNatKind(0) -> tt
14.03/15.00	isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	)
14.03/15.00	
14.03/15.00	Problem 1: 
14.03/15.00	
14.03/15.00	Dependency Pairs Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> U13#(isNat(V2))
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> U22#(isNat(V1))
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> U33#(isNat(V2))
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U41#(tt,N) -> N
14.03/15.00	 U51#(tt,M,N) -> PLUS(N,M)
14.03/15.00	 U51#(tt,M,N) -> M
14.03/15.00	 U51#(tt,M,N) -> N
14.03/15.00	 U71#(tt,M,N) -> PLUS(x(N,M),N)
14.03/15.00	 U71#(tt,M,N) -> X(N,M)
14.03/15.00	 U71#(tt,M,N) -> M
14.03/15.00	 U71#(tt,M,N) -> N
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 PLUS(N,0) -> U41#(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 PLUS(N,0) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 PLUS(N,0) -> ISNAT(N)
14.03/15.00	 PLUS(N,s(M)) -> U51#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 PLUS(N,s(M)) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
14.03/15.00	 PLUS(N,s(M)) -> AND(isNat(M),isNatKind(M))
14.03/15.00	 PLUS(N,s(M)) -> ISNAT(M)
14.03/15.00	 X(N,0) -> U61#(and(isNat(N),isNatKind(N)))
14.03/15.00	 X(N,0) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 X(N,0) -> ISNAT(N)
14.03/15.00	 X(N,s(M)) -> U71#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 X(N,s(M)) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
14.03/15.00	 X(N,s(M)) -> AND(isNat(M),isNatKind(M))
14.03/15.00	 X(N,s(M)) -> ISNAT(M)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding Rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	
14.03/15.00	Problem 1: 
14.03/15.00	
14.03/15.00	SCC Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> U13#(isNat(V2))
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> U22#(isNat(V1))
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> U33#(isNat(V2))
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U41#(tt,N) -> N
14.03/15.00	 U51#(tt,M,N) -> PLUS(N,M)
14.03/15.00	 U51#(tt,M,N) -> M
14.03/15.00	 U51#(tt,M,N) -> N
14.03/15.00	 U71#(tt,M,N) -> PLUS(x(N,M),N)
14.03/15.00	 U71#(tt,M,N) -> X(N,M)
14.03/15.00	 U71#(tt,M,N) -> M
14.03/15.00	 U71#(tt,M,N) -> N
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 PLUS(N,0) -> U41#(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 PLUS(N,0) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 PLUS(N,0) -> ISNAT(N)
14.03/15.00	 PLUS(N,s(M)) -> U51#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 PLUS(N,s(M)) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
14.03/15.00	 PLUS(N,s(M)) -> AND(isNat(M),isNatKind(M))
14.03/15.00	 PLUS(N,s(M)) -> ISNAT(M)
14.03/15.00	 X(N,0) -> U61#(and(isNat(N),isNatKind(N)))
14.03/15.00	 X(N,0) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 X(N,0) -> ISNAT(N)
14.03/15.00	 X(N,s(M)) -> U71#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 X(N,s(M)) -> AND(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N)))
14.03/15.00	 X(N,s(M)) -> AND(isNat(M),isNatKind(M))
14.03/15.00	 X(N,s(M)) -> ISNAT(M)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	->Strongly Connected Components:
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U51#(tt,M,N) -> PLUS(N,M)
14.03/15.00	 PLUS(N,s(M)) -> U51#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 Empty
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U71#(tt,M,N) -> X(N,M)
14.03/15.00	 X(N,s(M)) -> U71#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 Empty
14.03/15.00	
14.03/15.00	
14.03/15.00	The problem is decomposed in 3 subproblems.
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	Reduction Pairs Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	-> Usable rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->Interpretation type:
14.03/15.00	 Linear
14.03/15.00	->Coefficients:
14.03/15.00	 Natural Numbers
14.03/15.00	->Dimension:
14.03/15.00	 1
14.03/15.00	->Bound:
14.03/15.00	 2
14.03/15.00	->Interpretation:
14.03/15.00	 
14.03/15.00	[U11](X1,X2,X3) = 2.X1 + 1
14.03/15.00	[U12](X1,X2) = 1
14.03/15.00	[U13](X) = 1
14.03/15.00	[U21](X1,X2) = X1 + 2
14.03/15.00	[U22](X) = X
14.03/15.00	[U31](X1,X2,X3) = 2.X1 + 2
14.03/15.00	[U32](X1,X2) = 2
14.03/15.00	[U33](X) = 2
14.03/15.00	[U41](X1,X2) = 2.X2
14.03/15.00	[U51](X1,X2,X3) = 2.X3
14.03/15.00	[U61](X) = 0
14.03/15.00	[U71](X1,X2,X3) = 0
14.03/15.00	[and](X1,X2) = 2.X1 + 2.X2
14.03/15.00	[isNat](X) = 2
14.03/15.00	[isNatKind](X) = 0
14.03/15.00	[plus](X1,X2) = 2.X1
14.03/15.00	[x](X1,X2) = 0
14.03/15.00	[0] = 0
14.03/15.00	[s](X) = 0
14.03/15.00	[tt] = 0
14.03/15.00	[U11#](X1,X2,X3) = 2.X1 + 2
14.03/15.00	[U12#](X1,X2) = 2
14.03/15.00	[U21#](X1,X2) = 2
14.03/15.00	[U31#](X1,X2,X3) = 2
14.03/15.00	[U32#](X1,X2) = 2
14.03/15.00	[AND](X1,X2) = 2.X1 + X2
14.03/15.00	[ISNAT](X) = 2
14.03/15.00	[ISNATKIND](X) = 0
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	SCC Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	->Strongly Connected Components:
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	Reduction Pairs Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	-> Usable rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->Interpretation type:
14.03/15.00	 Linear
14.03/15.00	->Coefficients:
14.03/15.00	 Natural Numbers
14.03/15.00	->Dimension:
14.03/15.00	 1
14.03/15.00	->Bound:
14.03/15.00	 2
14.03/15.00	->Interpretation:
14.03/15.00	 
14.03/15.00	[U11](X1,X2,X3) = X1 + 1
14.03/15.00	[U12](X1,X2) = 1
14.03/15.00	[U13](X) = 1
14.03/15.00	[U21](X1,X2) = 2.X1 + 1
14.03/15.00	[U22](X) = 1
14.03/15.00	[U31](X1,X2,X3) = 2.X1 + X2 + 2
14.03/15.00	[U32](X1,X2) = X1
14.03/15.00	[U33](X) = 0
14.03/15.00	[U41](X1,X2) = X2
14.03/15.00	[U51](X1,X2,X3) = 0
14.03/15.00	[U61](X) = 2.X
14.03/15.00	[U71](X1,X2,X3) = 2.X3 + 2
14.03/15.00	[and](X1,X2) = X1 + 2.X2
14.03/15.00	[isNat](X) = X + 1
14.03/15.00	[isNatKind](X) = 0
14.03/15.00	[plus](X1,X2) = X1
14.03/15.00	[x](X1,X2) = 2.X1 + 2
14.03/15.00	[0] = 0
14.03/15.00	[s](X) = 0
14.03/15.00	[tt] = 0
14.03/15.00	[U11#](X1,X2,X3) = 2.X1 + 1
14.03/15.00	[U12#](X1,X2) = 1
14.03/15.00	[U21#](X1,X2) = X1 + 1
14.03/15.00	[U31#](X1,X2,X3) = 1
14.03/15.00	[U32#](X1,X2) = 1
14.03/15.00	[AND](X1,X2) = X1 + 2.X2
14.03/15.00	[ISNAT](X) = 1
14.03/15.00	[ISNATKIND](X) = 0
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	SCC Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	->Strongly Connected Components:
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	Reduction Pairs Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	-> Usable rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->Interpretation type:
14.03/15.00	 Linear
14.03/15.00	->Coefficients:
14.03/15.00	 Natural Numbers
14.03/15.00	->Dimension:
14.03/15.00	 1
14.03/15.00	->Bound:
14.03/15.00	 2
14.03/15.00	->Interpretation:
14.03/15.00	 
14.03/15.00	[U11](X1,X2,X3) = 2.X1 + 2
14.03/15.00	[U12](X1,X2) = 1
14.03/15.00	[U13](X) = 1
14.03/15.00	[U21](X1,X2) = 2
14.03/15.00	[U22](X) = X
14.03/15.00	[U31](X1,X2,X3) = 2.X1 + 2
14.03/15.00	[U32](X1,X2) = X1
14.03/15.00	[U33](X) = 0
14.03/15.00	[U41](X1,X2) = 2.X2
14.03/15.00	[U51](X1,X2,X3) = 2.X3
14.03/15.00	[U61](X) = 0
14.03/15.00	[U71](X1,X2,X3) = 0
14.03/15.00	[and](X1,X2) = 2.X1 + 2.X2
14.03/15.00	[isNat](X) = 2
14.03/15.00	[isNatKind](X) = 0
14.03/15.00	[plus](X1,X2) = 2.X1
14.03/15.00	[x](X1,X2) = 0
14.03/15.00	[0] = 0
14.03/15.00	[s](X) = 0
14.03/15.00	[tt] = 0
14.03/15.00	[U11#](X1,X2,X3) = 2.X1 + 2
14.03/15.00	[U12#](X1,X2) = 2
14.03/15.00	[U21#](X1,X2) = 2.X1 + 2
14.03/15.00	[U31#](X1,X2,X3) = 2.X1 + 2
14.03/15.00	[U32#](X1,X2) = 2
14.03/15.00	[AND](X1,X2) = 2.X1 + X2
14.03/15.00	[ISNAT](X) = 2
14.03/15.00	[ISNATKIND](X) = 0
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	SCC Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	->Strongly Connected Components:
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	Reduction Pairs Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	-> Usable rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->Interpretation type:
14.03/15.00	 Linear
14.03/15.00	->Coefficients:
14.03/15.00	 Natural Numbers
14.03/15.00	->Dimension:
14.03/15.00	 1
14.03/15.00	->Bound:
14.03/15.00	 2
14.03/15.00	->Interpretation:
14.03/15.00	 
14.03/15.00	[U11](X1,X2,X3) = X1 + 2.X2 + 2
14.03/15.00	[U12](X1,X2) = 2
14.03/15.00	[U13](X) = 2
14.03/15.00	[U21](X1,X2) = 2.X1 + 2.X2 + 2
14.03/15.00	[U22](X) = X
14.03/15.00	[U31](X1,X2,X3) = X2 + 2.X3
14.03/15.00	[U32](X1,X2) = 0
14.03/15.00	[U33](X) = 0
14.03/15.00	[U41](X1,X2) = X2
14.03/15.00	[U51](X1,X2,X3) = X3
14.03/15.00	[U61](X) = 1
14.03/15.00	[U71](X1,X2,X3) = 2.X2 + 2.X3 + 2
14.03/15.00	[and](X1,X2) = 2.X1 + X2
14.03/15.00	[isNat](X) = 2.X + 2
14.03/15.00	[isNatKind](X) = 0
14.03/15.00	[plus](X1,X2) = X1
14.03/15.00	[x](X1,X2) = 2.X1 + 2.X2 + 2
14.03/15.00	[0] = 0
14.03/15.00	[s](X) = X
14.03/15.00	[tt] = 0
14.03/15.00	[U11#](X1,X2,X3) = 2
14.03/15.00	[U12#](X1,X2) = 2
14.03/15.00	[U21#](X1,X2) = 2
14.03/15.00	[U31#](X1,X2,X3) = 2
14.03/15.00	[U32#](X1,X2) = 2
14.03/15.00	[AND](X1,X2) = 2.X1 + X2
14.03/15.00	[ISNAT](X) = 2
14.03/15.00	[ISNATKIND](X) = 0
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	SCC Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	->Strongly Connected Components:
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	Reduction Pairs Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNAT(s(V1)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	-> Usable rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->Interpretation type:
14.03/15.00	 Linear
14.03/15.00	->Coefficients:
14.03/15.00	 Natural Numbers
14.03/15.00	->Dimension:
14.03/15.00	 1
14.03/15.00	->Bound:
14.03/15.00	 2
14.03/15.00	->Interpretation:
14.03/15.00	 
14.03/15.00	[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2
14.03/15.00	[U12](X1,X2) = X1
14.03/15.00	[U13](X) = 0
14.03/15.00	[U21](X1,X2) = X1 + 2
14.03/15.00	[U22](X) = 2
14.03/15.00	[U31](X1,X2,X3) = X1 + X2 + 2.X3 + 2
14.03/15.00	[U32](X1,X2) = 2.X2 + 2
14.03/15.00	[U33](X) = 2
14.03/15.00	[U41](X1,X2) = X2
14.03/15.00	[U51](X1,X2,X3) = X3
14.03/15.00	[U61](X) = 2
14.03/15.00	[U71](X1,X2,X3) = 2.X2 + 2.X3 + 2
14.03/15.00	[and](X1,X2) = 2.X1 + 2.X2
14.03/15.00	[isNat](X) = 2.X + 2
14.03/15.00	[isNatKind](X) = 0
14.03/15.00	[plus](X1,X2) = X1
14.03/15.00	[x](X1,X2) = 2.X1 + 2.X2 + 2
14.03/15.00	[0] = 2
14.03/15.00	[s](X) = X
14.03/15.00	[tt] = 0
14.03/15.00	[U11#](X1,X2,X3) = 2
14.03/15.00	[U12#](X1,X2) = 2
14.03/15.00	[U21#](X1,X2) = X1 + 2
14.03/15.00	[U31#](X1,X2,X3) = 2.X1 + 2
14.03/15.00	[U32#](X1,X2) = 2
14.03/15.00	[AND](X1,X2) = 2.X1 + 2.X2
14.03/15.00	[ISNAT](X) = 2
14.03/15.00	[ISNATKIND](X) = 0
14.03/15.00	
14.03/15.00	Problem 1.1: 
14.03/15.00	
14.03/15.00	SCC Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 and(isNat(N),isNatKind(N)) -> ISNAT(N)
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	->Strongly Connected Components:
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 Empty
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 AND(tt,X) -> X
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.00	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.00	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.00	 isNatKind(M) -> ISNATKIND(M)
14.03/15.00	 isNatKind(N) -> ISNATKIND(N)
14.03/15.00	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.00	
14.03/15.00	
14.03/15.00	The problem is decomposed in 2 subproblems.
14.03/15.00	
14.03/15.00	Problem 1.1.1: 
14.03/15.00	
14.03/15.00	Reduction Pairs Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> U12#(isNat(V1),V2)
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 Empty
14.03/15.00	-> Usable rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->Interpretation type:
14.03/15.00	 Simple mixed
14.03/15.00	->Coefficients:
14.03/15.00	 Natural Numbers
14.03/15.00	->Dimension:
14.03/15.00	 1
14.03/15.00	->Bound:
14.03/15.00	 2
14.03/15.00	->Interpretation:
14.03/15.00	 
14.03/15.00	[U11](X1,X2,X3) = 2.X1.X2.X3 + X1.X3 + 2.X2.X3 + 2.X1 + 2.X2 + 2.X3
14.03/15.00	[U12](X1,X2) = X2
14.03/15.00	[U13](X) = 0
14.03/15.00	[U21](X1,X2) = 2.X1.X2 + 2.X2 + 2
14.03/15.00	[U22](X) = 2
14.03/15.00	[U31](X1,X2,X3) = X1.X2.X3 + 2.X2.X3 + 2.X1 + 2.X3 + 2
14.03/15.00	[U32](X1,X2) = 2.X2 + 2
14.03/15.00	[U33](X) = 0
14.03/15.00	[U41](X1,X2) = X2
14.03/15.00	[U51](X1,X2,X3) = 2.X1.X2.X3 + 2.X1.X2 + X2 + X3 + 2
14.03/15.00	[U61](X) = 2
14.03/15.00	[U71](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + 2.X3 + 2
14.03/15.00	[and](X1,X2) = 2.X2
14.03/15.00	[isNat](X) = 2.X.X + 2
14.03/15.00	[isNatKind](X) = 0
14.03/15.00	[plus](X1,X2) = X1 + X2 + 1
14.03/15.00	[x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1
14.03/15.00	[0] = 1
14.03/15.00	[s](X) = X + 1
14.03/15.00	[tt] = 0
14.03/15.00	[U11#](X1,X2,X3) = 2.X2 + 2.X3 + 2
14.03/15.00	[U12#](X1,X2) = 2.X2 + 1
14.03/15.00	[U21#](X1,X2) = 2.X2 + 1
14.03/15.00	[U31#](X1,X2,X3) = 2.X1.X2.X3 + X2.X3 + 2.X1 + 2.X2 + 2.X3 + 2
14.03/15.00	[U32#](X1,X2) = 2.X2 + 2
14.03/15.00	[ISNAT](X) = 2.X + 1
14.03/15.00	
14.03/15.00	Problem 1.1.1: 
14.03/15.00	
14.03/15.00	SCC Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U12#(tt,V2) -> ISNAT(V2)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 Empty
14.03/15.00	->Strongly Connected Components:
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	->->-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->->-> Unhiding rules:
14.03/15.00	 Empty
14.03/15.00	
14.03/15.00	Problem 1.1.1: 
14.03/15.00	
14.03/15.00	Reduction Pairs Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> U32#(isNat(V1),V2)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 Empty
14.03/15.00	-> Usable rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	->Interpretation type:
14.03/15.00	 Linear
14.03/15.00	->Coefficients:
14.03/15.00	 Natural Numbers
14.03/15.00	->Dimension:
14.03/15.00	 1
14.03/15.00	->Bound:
14.03/15.00	 2
14.03/15.00	->Interpretation:
14.03/15.00	 
14.03/15.00	[U11](X1,X2,X3) = X1
14.03/15.00	[U12](X1,X2) = 2
14.03/15.00	[U13](X) = 2
14.03/15.00	[U21](X1,X2) = X1
14.03/15.00	[U22](X) = 2
14.03/15.00	[U31](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
14.03/15.00	[U32](X1,X2) = 2.X1 + 2.X2
14.03/15.00	[U33](X) = 2.X
14.03/15.00	[U41](X1,X2) = X2
14.03/15.00	[U51](X1,X2,X3) = X3
14.03/15.00	[U61](X) = 2.X + 2
14.03/15.00	[U71](X1,X2,X3) = X1 + 2.X2 + 2.X3
14.03/15.00	[and](X1,X2) = X2
14.03/15.00	[isNat](X) = X + 2
14.03/15.00	[isNatKind](X) = 2
14.03/15.00	[plus](X1,X2) = X1
14.03/15.00	[x](X1,X2) = 2.X1 + 2.X2 + 2
14.03/15.00	[0] = 2
14.03/15.00	[s](X) = X
14.03/15.00	[tt] = 2
14.03/15.00	[U11#](X1,X2,X3) = 2.X2 + 1
14.03/15.00	[U21#](X1,X2) = 2.X2 + 1
14.03/15.00	[U31#](X1,X2,X3) = 2.X2 + 2.X3 + 2
14.03/15.00	[U32#](X1,X2) = 2.X2 + 1
14.03/15.00	[ISNAT](X) = 2.X + 1
14.03/15.00	
14.03/15.00	Problem 1.1.1: 
14.03/15.00	
14.03/15.00	SCC Processor:
14.03/15.00	-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.00	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.00	 U32#(tt,V2) -> ISNAT(V2)
14.03/15.00	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.00	-> Rules:
14.03/15.00	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.00	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.00	 U13(tt) -> tt
14.03/15.00	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.00	 U22(tt) -> tt
14.03/15.00	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.00	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.00	 U33(tt) -> tt
14.03/15.00	 U41(tt,N) -> N
14.03/15.00	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.00	 U61(tt) -> 0
14.03/15.00	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.00	 and(tt,X) -> X
14.03/15.00	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.00	 isNat(0) -> tt
14.03/15.00	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.00	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.00	 isNatKind(0) -> tt
14.03/15.00	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.00	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.00	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.00	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.00	-> Unhiding rules:
14.03/15.00	 Empty
14.03/15.00	->Strongly Connected Components:
14.03/15.00	->->Cycle:
14.03/15.00	->->-> Pairs:
14.03/15.00	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.01	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.01	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.01	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.01	->->-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	->->-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	
14.03/15.01	Problem 1.1.1: 
14.03/15.01	
14.03/15.01	SubNColl Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.01	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.01	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.01	 ISNAT(plus(V1,V2)) -> U11#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 ISNAT(x(V1,V2)) -> U31#(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	->Projection:
14.03/15.01	 pi(U11#) = 2
14.03/15.01	 pi(U21#) = 2
14.03/15.01	 pi(U31#) = 2
14.03/15.01	 pi(ISNAT) = 1
14.03/15.01	
14.03/15.01	Problem 1.1.1: 
14.03/15.01	
14.03/15.01	SCC Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 U11#(tt,V1,V2) -> ISNAT(V1)
14.03/15.01	 U21#(tt,V1) -> ISNAT(V1)
14.03/15.01	 U31#(tt,V1,V2) -> ISNAT(V1)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	->Strongly Connected Components:
14.03/15.01	 There is no strongly connected component
14.03/15.01	
14.03/15.01	The problem is finite.
14.03/15.01	
14.03/15.01	Problem 1.1.2: 
14.03/15.01	
14.03/15.01	Reduction Pairs Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 AND(tt,X) -> X
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 and(isNat(N),isNatKind(N)) -> AND(isNat(N),isNatKind(N))
14.03/15.01	 isNatKind(M) -> ISNATKIND(M)
14.03/15.01	 isNatKind(N) -> ISNATKIND(N)
14.03/15.01	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.01	-> Usable rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	->Interpretation type:
14.03/15.01	 Linear
14.03/15.01	->Coefficients:
14.03/15.01	 Natural Numbers
14.03/15.01	->Dimension:
14.03/15.01	 1
14.03/15.01	->Bound:
14.03/15.01	 2
14.03/15.01	->Interpretation:
14.03/15.01	 
14.03/15.01	[U11](X1,X2,X3) = X1 + 1
14.03/15.01	[U12](X1,X2) = 1
14.03/15.01	[U13](X) = 0
14.03/15.01	[U21](X1,X2) = X1 + 1
14.03/15.01	[U22](X) = 1
14.03/15.01	[U31](X1,X2,X3) = 2.X3 + 2
14.03/15.01	[U32](X1,X2) = 2.X2 + 2
14.03/15.01	[U33](X) = 2.X
14.03/15.01	[U41](X1,X2) = X2
14.03/15.01	[U51](X1,X2,X3) = X3
14.03/15.01	[U61](X) = 2
14.03/15.01	[U71](X1,X2,X3) = 2.X2 + 2
14.03/15.01	[and](X1,X2) = X1 + X2
14.03/15.01	[isNat](X) = X + 1
14.03/15.01	[isNatKind](X) = 0
14.03/15.01	[plus](X1,X2) = X1
14.03/15.01	[x](X1,X2) = 2.X2 + 2
14.03/15.01	[0] = 1
14.03/15.01	[s](X) = X
14.03/15.01	[tt] = 0
14.03/15.01	[AND](X1,X2) = X2
14.03/15.01	[ISNATKIND](X) = 0
14.03/15.01	
14.03/15.01	Problem 1.1.2: 
14.03/15.01	
14.03/15.01	SCC Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 AND(tt,X) -> X
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 isNatKind(M) -> ISNATKIND(M)
14.03/15.01	 isNatKind(N) -> ISNATKIND(N)
14.03/15.01	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.01	->Strongly Connected Components:
14.03/15.01	->->Cycle:
14.03/15.01	->->-> Pairs:
14.03/15.01	 AND(tt,X) -> X
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.01	->->-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	->->-> Unhiding rules:
14.03/15.01	 isNatKind(M) -> ISNATKIND(M)
14.03/15.01	 isNatKind(N) -> ISNATKIND(N)
14.03/15.01	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.01	
14.03/15.01	Problem 1.1.2: 
14.03/15.01	
14.03/15.01	Reduction Pairs Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 AND(tt,X) -> X
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 isNatKind(M) -> ISNATKIND(M)
14.03/15.01	 isNatKind(N) -> ISNATKIND(N)
14.03/15.01	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.01	-> Usable rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	->Interpretation type:
14.03/15.01	 Simple mixed
14.03/15.01	->Coefficients:
14.03/15.01	 Natural Numbers
14.03/15.01	->Dimension:
14.03/15.01	 1
14.03/15.01	->Bound:
14.03/15.01	 2
14.03/15.01	->Interpretation:
14.03/15.01	 
14.03/15.01	[U11](X1,X2,X3) = 2.X2.X3 + X1 + 2.X2 + X3 + 2
14.03/15.01	[U12](X1,X2) = 1
14.03/15.01	[U13](X) = 0
14.03/15.01	[U21](X1,X2) = X1
14.03/15.01	[U22](X) = 0
14.03/15.01	[U31](X1,X2,X3) = 2.X1.X2 + 2.X2.X3 + 2.X2 + 2
14.03/15.01	[U32](X1,X2) = 2
14.03/15.01	[U33](X) = 2
14.03/15.01	[U41](X1,X2) = X2 + 2
14.03/15.01	[U51](X1,X2,X3) = X2 + X3 + 2
14.03/15.01	[U61](X) = 2
14.03/15.01	[U71](X1,X2,X3) = 2.X2.X3 + 2.X2 + 2.X3 + 2
14.03/15.01	[and](X1,X2) = X2 + 2
14.03/15.01	[isNat](X) = 2.X.X + 2.X
14.03/15.01	[isNatKind](X) = 2.X.X
14.03/15.01	[plus](X1,X2) = X1 + X2 + 1
14.03/15.01	[x](X1,X2) = 2.X1.X2 + X1 + 2.X2 + 1
14.03/15.01	[0] = 2
14.03/15.01	[s](X) = X + 1
14.03/15.01	[tt] = 0
14.03/15.01	[AND](X1,X2) = X2 + 1
14.03/15.01	[ISNATKIND](X) = 2.X.X
14.03/15.01	
14.03/15.01	Problem 1.1.2: 
14.03/15.01	
14.03/15.01	SCC Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(x(V1,V2)) -> AND(isNatKind(V1),isNatKind(V2))
14.03/15.01	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 isNatKind(M) -> ISNATKIND(M)
14.03/15.01	 isNatKind(N) -> ISNATKIND(N)
14.03/15.01	 isNatKind(V2) -> ISNATKIND(V2)
14.03/15.01	->Strongly Connected Components:
14.03/15.01	->->Cycle:
14.03/15.01	->->-> Pairs:
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.01	->->-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	->->-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	
14.03/15.01	Problem 1.1.2: 
14.03/15.01	
14.03/15.01	SubNColl Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
14.03/15.01	 ISNATKIND(s(V1)) -> ISNATKIND(V1)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	->Projection:
14.03/15.01	 pi(ISNATKIND) = 1
14.03/15.01	
14.03/15.01	Problem 1.1.2: 
14.03/15.01	
14.03/15.01	Basic Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 Empty
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	-> Result:
14.03/15.01	 Set P is empty
14.03/15.01	
14.03/15.01	The problem is finite.
14.03/15.01	
14.03/15.01	Problem 1.2: 
14.03/15.01	
14.03/15.01	SubNColl Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 U51#(tt,M,N) -> PLUS(N,M)
14.03/15.01	 PLUS(N,s(M)) -> U51#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	->Projection:
14.03/15.01	 pi(U51#) = 2
14.03/15.01	 pi(PLUS) = 2
14.03/15.01	
14.03/15.01	Problem 1.2: 
14.03/15.01	
14.03/15.01	SCC Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 U51#(tt,M,N) -> PLUS(N,M)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	->Strongly Connected Components:
14.03/15.01	 There is no strongly connected component
14.03/15.01	
14.03/15.01	The problem is finite.
14.03/15.01	
14.03/15.01	Problem 1.3: 
14.03/15.01	
14.03/15.01	SubNColl Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 U71#(tt,M,N) -> X(N,M)
14.03/15.01	 X(N,s(M)) -> U71#(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	->Projection:
14.03/15.01	 pi(U71#) = 2
14.03/15.01	 pi(X) = 2
14.03/15.01	
14.03/15.01	Problem 1.3: 
14.03/15.01	
14.03/15.01	SCC Processor:
14.03/15.01	-> Pairs:
14.03/15.01	 U71#(tt,M,N) -> X(N,M)
14.03/15.01	-> Rules:
14.03/15.01	 U11(tt,V1,V2) -> U12(isNat(V1),V2)
14.03/15.01	 U12(tt,V2) -> U13(isNat(V2))
14.03/15.01	 U13(tt) -> tt
14.03/15.01	 U21(tt,V1) -> U22(isNat(V1))
14.03/15.01	 U22(tt) -> tt
14.03/15.01	 U31(tt,V1,V2) -> U32(isNat(V1),V2)
14.03/15.01	 U32(tt,V2) -> U33(isNat(V2))
14.03/15.01	 U33(tt) -> tt
14.03/15.01	 U41(tt,N) -> N
14.03/15.01	 U51(tt,M,N) -> s(plus(N,M))
14.03/15.01	 U61(tt) -> 0
14.03/15.01	 U71(tt,M,N) -> plus(x(N,M),N)
14.03/15.01	 and(tt,X) -> X
14.03/15.01	 isNat(plus(V1,V2)) -> U11(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(x(V1,V2)) -> U31(and(isNatKind(V1),isNatKind(V2)),V1,V2)
14.03/15.01	 isNat(0) -> tt
14.03/15.01	 isNat(s(V1)) -> U21(isNatKind(V1),V1)
14.03/15.01	 isNatKind(plus(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(x(V1,V2)) -> and(isNatKind(V1),isNatKind(V2))
14.03/15.01	 isNatKind(0) -> tt
14.03/15.01	 isNatKind(s(V1)) -> isNatKind(V1)
14.03/15.01	 plus(N,0) -> U41(and(isNat(N),isNatKind(N)),N)
14.03/15.01	 plus(N,s(M)) -> U51(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	 x(N,0) -> U61(and(isNat(N),isNatKind(N)))
14.03/15.01	 x(N,s(M)) -> U71(and(and(isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N)
14.03/15.01	-> Unhiding rules:
14.03/15.01	 Empty
14.03/15.01	->Strongly Connected Components:
14.03/15.01	 There is no strongly connected component
14.03/15.01	
14.03/15.01	The problem is finite.
14.03/15.01	EOF