16.01/16.86	YES
16.01/16.86	
16.01/16.86	Problem 1: 
16.01/16.86	
16.01/16.86	(VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S)
16.01/16.86	(RULES
16.01/16.86	U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.86	U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.86	U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.86	U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.86	U13(active(X:S)) -> U13(X:S)
16.01/16.86	U13(mark(X:S)) -> U13(X:S)
16.01/16.86	U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.86	U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.86	U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.86	U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.86	U22(active(X:S)) -> U22(X:S)
16.01/16.86	U22(mark(X:S)) -> U22(X:S)
16.01/16.86	U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.86	U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.86	U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.86	U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.86	U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.86	active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.86	active(U13(tt)) -> mark(tt)
16.01/16.86	active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.86	active(U22(tt)) -> mark(tt)
16.01/16.86	active(U31(tt,N:S)) -> mark(N:S)
16.01/16.86	active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.86	active(and(tt,X:S)) -> mark(X:S)
16.01/16.86	active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.86	active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.86	active(isNat(0)) -> mark(tt)
16.01/16.86	active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.86	active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.86	active(isNatKind(0)) -> mark(tt)
16.01/16.86	active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.86	active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.86	and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.86	and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.86	and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.86	and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.86	isNat(active(X:S)) -> isNat(X:S)
16.01/16.86	isNat(mark(X:S)) -> isNat(X:S)
16.01/16.86	isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.86	isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.86	mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.86	mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.86	mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.86	mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.86	mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.86	mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.86	mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.86	mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.86	mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.86	mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.86	mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.86	mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.86	mark(0) -> active(0)
16.01/16.86	mark(tt) -> active(tt)
16.01/16.86	plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.86	plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.86	plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.86	plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.86	s(active(X:S)) -> s(X:S)
16.01/16.86	s(mark(X:S)) -> s(X:S)
16.01/16.86	) 
16.01/16.86	(STRATEGY INNERMOST)
16.01/16.86	
16.01/16.86	Problem 1: 
16.01/16.86	
16.01/16.86	Dependency Pairs Processor:
16.01/16.86	-> Pairs:
16.01/16.86	 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U12#(active(X1:S),X2:S) -> U12#(X1:S,X2:S)
16.01/16.86	 U12#(mark(X1:S),X2:S) -> U12#(X1:S,X2:S)
16.01/16.86	 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.86	 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.86	 U13#(active(X:S)) -> U13#(X:S)
16.01/16.86	 U13#(mark(X:S)) -> U13#(X:S)
16.01/16.86	 U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S)
16.01/16.86	 U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S)
16.01/16.86	 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.86	 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.86	 U22#(active(X:S)) -> U22#(X:S)
16.01/16.86	 U22#(mark(X:S)) -> U22#(X:S)
16.01/16.86	 U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S)
16.01/16.86	 U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S)
16.01/16.86	 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.86	 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.86	 U41#(active(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(mark(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 ACTIVE(U11(tt,V1:S,V2:S)) -> U12#(isNat(V1:S),V2:S)
16.01/16.86	 ACTIVE(U11(tt,V1:S,V2:S)) -> ISNAT(V1:S)
16.01/16.86	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.86	 ACTIVE(U12(tt,V2:S)) -> U13#(isNat(V2:S))
16.01/16.86	 ACTIVE(U12(tt,V2:S)) -> ISNAT(V2:S)
16.01/16.86	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.86	 ACTIVE(U13(tt)) -> MARK(tt)
16.01/16.86	 ACTIVE(U21(tt,V1:S)) -> U22#(isNat(V1:S))
16.01/16.86	 ACTIVE(U21(tt,V1:S)) -> ISNAT(V1:S)
16.01/16.86	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.86	 ACTIVE(U22(tt)) -> MARK(tt)
16.01/16.86	 ACTIVE(U31(tt,N:S)) -> MARK(N:S)
16.01/16.86	 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
16.01/16.86	 ACTIVE(U41(tt,M:S,N:S)) -> PLUS(N:S,M:S)
16.01/16.86	 ACTIVE(U41(tt,M:S,N:S)) -> S(plus(N:S,M:S))
16.01/16.86	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> U11#(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S)
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> AND(isNatKind(V1:S),isNatKind(V2:S))
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V1:S)
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V2:S)
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.86	 ACTIVE(isNat(s(V1:S))) -> U21#(isNatKind(V1:S),V1:S)
16.01/16.86	 ACTIVE(isNat(s(V1:S))) -> ISNATKIND(V1:S)
16.01/16.86	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.86	 ACTIVE(isNat(0)) -> MARK(tt)
16.01/16.86	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> AND(isNatKind(V1:S),isNatKind(V2:S))
16.01/16.86	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V1:S)
16.01/16.86	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V2:S)
16.01/16.86	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.86	 ACTIVE(isNatKind(s(V1:S))) -> ISNATKIND(V1:S)
16.01/16.86	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.86	 ACTIVE(isNatKind(0)) -> MARK(tt)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> U41#(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> AND(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S)))
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(M:S),isNatKind(M:S))
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(N:S),isNatKind(N:S))
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(N:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> ISNATKIND(M:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> ISNATKIND(N:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.86	 ACTIVE(plus(N:S,0)) -> U31#(and(isNat(N:S),isNatKind(N:S)),N:S)
16.01/16.86	 ACTIVE(plus(N:S,0)) -> AND(isNat(N:S),isNatKind(N:S))
16.01/16.86	 ACTIVE(plus(N:S,0)) -> ISNAT(N:S)
16.01/16.86	 ACTIVE(plus(N:S,0)) -> ISNATKIND(N:S)
16.01/16.86	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.86	 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
16.01/16.86	 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
16.01/16.86	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.01/16.86	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.01/16.86	 ISNAT(active(X:S)) -> ISNAT(X:S)
16.01/16.86	 ISNAT(mark(X:S)) -> ISNAT(X:S)
16.01/16.86	 ISNATKIND(active(X:S)) -> ISNATKIND(X:S)
16.01/16.86	 ISNATKIND(mark(X:S)) -> ISNATKIND(X:S)
16.01/16.86	 MARK(U11(X1:S,X2:S,X3:S)) -> U11#(mark(X1:S),X2:S,X3:S)
16.01/16.86	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.86	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.86	 MARK(U12(X1:S,X2:S)) -> U12#(mark(X1:S),X2:S)
16.01/16.86	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.86	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.86	 MARK(U13(X:S)) -> U13#(mark(X:S))
16.01/16.86	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.86	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.86	 MARK(U21(X1:S,X2:S)) -> U21#(mark(X1:S),X2:S)
16.01/16.86	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.86	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.86	 MARK(U22(X:S)) -> U22#(mark(X:S))
16.01/16.86	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.86	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.86	 MARK(U31(X1:S,X2:S)) -> U31#(mark(X1:S),X2:S)
16.01/16.86	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.86	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.86	 MARK(U41(X1:S,X2:S,X3:S)) -> U41#(mark(X1:S),X2:S,X3:S)
16.01/16.86	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.86	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.86	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.86	 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S)
16.01/16.86	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.86	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.86	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.86	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.86	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.86	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.86	 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S))
16.01/16.86	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.86	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.86	 MARK(s(X:S)) -> S(mark(X:S))
16.01/16.86	 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.01/16.86	 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.01/16.86	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.86	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.86	 S(active(X:S)) -> S(X:S)
16.01/16.86	 S(mark(X:S)) -> S(X:S)
16.01/16.86	-> Rules:
16.01/16.86	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.86	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.86	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.86	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.86	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.86	 U13(active(X:S)) -> U13(X:S)
16.01/16.86	 U13(mark(X:S)) -> U13(X:S)
16.01/16.86	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.86	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.86	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.86	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.86	 U22(active(X:S)) -> U22(X:S)
16.01/16.86	 U22(mark(X:S)) -> U22(X:S)
16.01/16.86	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.86	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.86	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.86	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.86	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.86	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.86	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.86	 active(U13(tt)) -> mark(tt)
16.01/16.86	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.86	 active(U22(tt)) -> mark(tt)
16.01/16.86	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.86	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.86	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.86	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.86	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.86	 active(isNat(0)) -> mark(tt)
16.01/16.86	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.86	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.86	 active(isNatKind(0)) -> mark(tt)
16.01/16.86	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.86	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.86	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.86	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.86	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.86	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.86	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.86	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.86	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.86	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.86	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.86	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.86	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.86	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.86	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.86	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.86	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.86	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.86	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.86	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.86	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.86	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.86	 mark(0) -> active(0)
16.01/16.86	 mark(tt) -> active(tt)
16.01/16.86	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.86	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.86	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.86	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.86	 s(active(X:S)) -> s(X:S)
16.01/16.86	 s(mark(X:S)) -> s(X:S)
16.01/16.86	
16.01/16.86	Problem 1: 
16.01/16.86	
16.01/16.86	SCC Processor:
16.01/16.86	-> Pairs:
16.01/16.86	 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.86	 U12#(active(X1:S),X2:S) -> U12#(X1:S,X2:S)
16.01/16.86	 U12#(mark(X1:S),X2:S) -> U12#(X1:S,X2:S)
16.01/16.86	 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.86	 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.86	 U13#(active(X:S)) -> U13#(X:S)
16.01/16.86	 U13#(mark(X:S)) -> U13#(X:S)
16.01/16.86	 U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S)
16.01/16.86	 U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S)
16.01/16.86	 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.86	 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.86	 U22#(active(X:S)) -> U22#(X:S)
16.01/16.86	 U22#(mark(X:S)) -> U22#(X:S)
16.01/16.86	 U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S)
16.01/16.86	 U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S)
16.01/16.86	 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.86	 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.86	 U41#(active(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(mark(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.86	 ACTIVE(U11(tt,V1:S,V2:S)) -> U12#(isNat(V1:S),V2:S)
16.01/16.86	 ACTIVE(U11(tt,V1:S,V2:S)) -> ISNAT(V1:S)
16.01/16.86	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.86	 ACTIVE(U12(tt,V2:S)) -> U13#(isNat(V2:S))
16.01/16.86	 ACTIVE(U12(tt,V2:S)) -> ISNAT(V2:S)
16.01/16.86	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.86	 ACTIVE(U13(tt)) -> MARK(tt)
16.01/16.86	 ACTIVE(U21(tt,V1:S)) -> U22#(isNat(V1:S))
16.01/16.86	 ACTIVE(U21(tt,V1:S)) -> ISNAT(V1:S)
16.01/16.86	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.86	 ACTIVE(U22(tt)) -> MARK(tt)
16.01/16.86	 ACTIVE(U31(tt,N:S)) -> MARK(N:S)
16.01/16.86	 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
16.01/16.86	 ACTIVE(U41(tt,M:S,N:S)) -> PLUS(N:S,M:S)
16.01/16.86	 ACTIVE(U41(tt,M:S,N:S)) -> S(plus(N:S,M:S))
16.01/16.86	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> U11#(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S)
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> AND(isNatKind(V1:S),isNatKind(V2:S))
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V1:S)
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V2:S)
16.01/16.86	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.86	 ACTIVE(isNat(s(V1:S))) -> U21#(isNatKind(V1:S),V1:S)
16.01/16.86	 ACTIVE(isNat(s(V1:S))) -> ISNATKIND(V1:S)
16.01/16.86	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.86	 ACTIVE(isNat(0)) -> MARK(tt)
16.01/16.86	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> AND(isNatKind(V1:S),isNatKind(V2:S))
16.01/16.86	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V1:S)
16.01/16.86	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V2:S)
16.01/16.86	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.86	 ACTIVE(isNatKind(s(V1:S))) -> ISNATKIND(V1:S)
16.01/16.86	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.86	 ACTIVE(isNatKind(0)) -> MARK(tt)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> U41#(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> AND(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S)))
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(M:S),isNatKind(M:S))
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(N:S),isNatKind(N:S))
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(N:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> ISNATKIND(M:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> ISNATKIND(N:S)
16.01/16.86	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.86	 ACTIVE(plus(N:S,0)) -> U31#(and(isNat(N:S),isNatKind(N:S)),N:S)
16.01/16.86	 ACTIVE(plus(N:S,0)) -> AND(isNat(N:S),isNatKind(N:S))
16.01/16.86	 ACTIVE(plus(N:S,0)) -> ISNAT(N:S)
16.01/16.86	 ACTIVE(plus(N:S,0)) -> ISNATKIND(N:S)
16.01/16.86	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.86	 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
16.01/16.86	 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
16.01/16.86	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.01/16.86	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.01/16.86	 ISNAT(active(X:S)) -> ISNAT(X:S)
16.01/16.86	 ISNAT(mark(X:S)) -> ISNAT(X:S)
16.01/16.86	 ISNATKIND(active(X:S)) -> ISNATKIND(X:S)
16.01/16.86	 ISNATKIND(mark(X:S)) -> ISNATKIND(X:S)
16.01/16.86	 MARK(U11(X1:S,X2:S,X3:S)) -> U11#(mark(X1:S),X2:S,X3:S)
16.01/16.86	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.86	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.86	 MARK(U12(X1:S,X2:S)) -> U12#(mark(X1:S),X2:S)
16.01/16.86	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.86	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.86	 MARK(U13(X:S)) -> U13#(mark(X:S))
16.01/16.86	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.86	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.86	 MARK(U21(X1:S,X2:S)) -> U21#(mark(X1:S),X2:S)
16.01/16.86	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.86	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.86	 MARK(U22(X:S)) -> U22#(mark(X:S))
16.01/16.86	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.86	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.86	 MARK(U31(X1:S,X2:S)) -> U31#(mark(X1:S),X2:S)
16.01/16.86	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.86	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.86	 MARK(U41(X1:S,X2:S,X3:S)) -> U41#(mark(X1:S),X2:S,X3:S)
16.01/16.86	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.87	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.87	 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S)
16.01/16.87	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.87	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.87	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.87	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.87	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.87	 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S))
16.01/16.87	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.87	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.87	 MARK(s(X:S)) -> S(mark(X:S))
16.01/16.87	 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	 S(active(X:S)) -> S(X:S)
16.01/16.87	 S(mark(X:S)) -> S(X:S)
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Strongly Connected Components:
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 S(active(X:S)) -> S(X:S)
16.01/16.87	 S(mark(X:S)) -> S(X:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 ISNATKIND(active(X:S)) -> ISNATKIND(X:S)
16.01/16.87	 ISNATKIND(mark(X:S)) -> ISNATKIND(X:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 ISNAT(active(X:S)) -> ISNAT(X:S)
16.01/16.87	 ISNAT(mark(X:S)) -> ISNAT(X:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
16.01/16.87	 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
16.01/16.87	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.01/16.87	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 U41#(active(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.87	 U41#(mark(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.87	 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.87	 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.87	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.87	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S)
16.01/16.87	 U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S)
16.01/16.87	 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.87	 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 U22#(active(X:S)) -> U22#(X:S)
16.01/16.87	 U22#(mark(X:S)) -> U22#(X:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S)
16.01/16.87	 U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S)
16.01/16.87	 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.87	 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 U13#(active(X:S)) -> U13#(X:S)
16.01/16.87	 U13#(mark(X:S)) -> U13#(X:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 U12#(active(X1:S),X2:S) -> U12#(X1:S,X2:S)
16.01/16.87	 U12#(mark(X1:S),X2:S) -> U12#(X1:S,X2:S)
16.01/16.87	 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.87	 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.87	 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.87	 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.87	 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.87	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.87	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.87	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.87	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.87	 ACTIVE(U31(tt,N:S)) -> MARK(N:S)
16.01/16.87	 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
16.01/16.87	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.87	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.87	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.87	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.87	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.87	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.87	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.87	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.87	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.87	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.87	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.87	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.87	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.87	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.87	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.87	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.87	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.87	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.87	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.87	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.87	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.87	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	
16.01/16.87	
16.01/16.87	The problem is decomposed in 13 subproblems.
16.01/16.87	
16.01/16.87	Problem 1.1: 
16.01/16.87	
16.01/16.87	Subterm Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 S(active(X:S)) -> S(X:S)
16.01/16.87	 S(mark(X:S)) -> S(X:S)
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Projection:
16.01/16.87	 pi(S) = 1
16.01/16.87	
16.01/16.87	Problem 1.1: 
16.01/16.87	
16.01/16.87	SCC Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 Empty
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Strongly Connected Components:
16.01/16.87	 There is no strongly connected component
16.01/16.87	
16.01/16.87	The problem is finite.
16.01/16.87	
16.01/16.87	Problem 1.2: 
16.01/16.87	
16.01/16.87	Subterm Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Projection:
16.01/16.87	 pi(PLUS) = 1
16.01/16.87	
16.01/16.87	Problem 1.2: 
16.01/16.87	
16.01/16.87	SCC Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Strongly Connected Components:
16.01/16.87	->->Cycle:
16.01/16.87	->->-> Pairs:
16.01/16.87	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	->->-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	
16.01/16.87	Problem 1.2: 
16.01/16.87	
16.01/16.87	Subterm Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Projection:
16.01/16.87	 pi(PLUS) = 2
16.01/16.87	
16.01/16.87	Problem 1.2: 
16.01/16.87	
16.01/16.87	SCC Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 Empty
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Strongly Connected Components:
16.01/16.87	 There is no strongly connected component
16.01/16.87	
16.01/16.87	The problem is finite.
16.01/16.87	
16.01/16.87	Problem 1.3: 
16.01/16.87	
16.01/16.87	Subterm Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 ISNATKIND(active(X:S)) -> ISNATKIND(X:S)
16.01/16.87	 ISNATKIND(mark(X:S)) -> ISNATKIND(X:S)
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Projection:
16.01/16.87	 pi(ISNATKIND) = 1
16.01/16.87	
16.01/16.87	Problem 1.3: 
16.01/16.87	
16.01/16.87	SCC Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 Empty
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Strongly Connected Components:
16.01/16.87	 There is no strongly connected component
16.01/16.87	
16.01/16.87	The problem is finite.
16.01/16.87	
16.01/16.87	Problem 1.4: 
16.01/16.87	
16.01/16.87	Subterm Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 ISNAT(active(X:S)) -> ISNAT(X:S)
16.01/16.87	 ISNAT(mark(X:S)) -> ISNAT(X:S)
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Projection:
16.01/16.87	 pi(ISNAT) = 1
16.01/16.87	
16.01/16.87	Problem 1.4: 
16.01/16.87	
16.01/16.87	SCC Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 Empty
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.87	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.87	 U13(active(X:S)) -> U13(X:S)
16.01/16.87	 U13(mark(X:S)) -> U13(X:S)
16.01/16.87	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.87	 U22(active(X:S)) -> U22(X:S)
16.01/16.87	 U22(mark(X:S)) -> U22(X:S)
16.01/16.87	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.87	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.87	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.87	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.87	 active(U13(tt)) -> mark(tt)
16.01/16.87	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.87	 active(U22(tt)) -> mark(tt)
16.01/16.87	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.87	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.87	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.87	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.87	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.87	 active(isNat(0)) -> mark(tt)
16.01/16.87	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.87	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.87	 active(isNatKind(0)) -> mark(tt)
16.01/16.87	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.87	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.87	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.87	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.87	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.87	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.87	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.87	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.87	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.87	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.87	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.87	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.87	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.87	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.87	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.87	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.87	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.87	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.87	 mark(0) -> active(0)
16.01/16.87	 mark(tt) -> active(tt)
16.01/16.87	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.87	 s(active(X:S)) -> s(X:S)
16.01/16.87	 s(mark(X:S)) -> s(X:S)
16.01/16.87	->Strongly Connected Components:
16.01/16.87	 There is no strongly connected component
16.01/16.87	
16.01/16.87	The problem is finite.
16.01/16.87	
16.01/16.87	Problem 1.5: 
16.01/16.87	
16.01/16.87	Subterm Processor:
16.01/16.87	-> Pairs:
16.01/16.87	 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
16.01/16.87	 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
16.01/16.87	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.01/16.87	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.01/16.87	-> Rules:
16.01/16.87	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(AND) = 1
16.01/16.88	
16.01/16.88	Problem 1.5: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.01/16.88	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	->->Cycle:
16.01/16.88	->->-> Pairs:
16.01/16.88	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.01/16.88	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.01/16.88	->->-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	
16.01/16.88	Problem 1.5: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
16.01/16.88	 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(AND) = 2
16.01/16.88	
16.01/16.88	Problem 1.5: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 Empty
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	 There is no strongly connected component
16.01/16.88	
16.01/16.88	The problem is finite.
16.01/16.88	
16.01/16.88	Problem 1.6: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U41#(active(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(mark(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(U41#) = 1
16.01/16.88	
16.01/16.88	Problem 1.6: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	->->Cycle:
16.01/16.88	->->-> Pairs:
16.01/16.88	 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	->->-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	
16.01/16.88	Problem 1.6: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(U41#) = 2
16.01/16.88	
16.01/16.88	Problem 1.6: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	->->Cycle:
16.01/16.88	->->-> Pairs:
16.01/16.88	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	->->-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	
16.01/16.88	Problem 1.6: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(U41#) = 3
16.01/16.88	
16.01/16.88	Problem 1.6: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 Empty
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	 There is no strongly connected component
16.01/16.88	
16.01/16.88	The problem is finite.
16.01/16.88	
16.01/16.88	Problem 1.7: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S)
16.01/16.88	 U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S)
16.01/16.88	 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.88	 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(U31#) = 1
16.01/16.88	
16.01/16.88	Problem 1.7: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.88	 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	->->Cycle:
16.01/16.88	->->-> Pairs:
16.01/16.88	 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.88	 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.88	->->-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	
16.01/16.88	Problem 1.7: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.88	 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(U31#) = 2
16.01/16.88	
16.01/16.88	Problem 1.7: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 Empty
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	 There is no strongly connected component
16.01/16.88	
16.01/16.88	The problem is finite.
16.01/16.88	
16.01/16.88	Problem 1.8: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U22#(active(X:S)) -> U22#(X:S)
16.01/16.88	 U22#(mark(X:S)) -> U22#(X:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(U22#) = 1
16.01/16.88	
16.01/16.88	Problem 1.8: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 Empty
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	 There is no strongly connected component
16.01/16.88	
16.01/16.88	The problem is finite.
16.01/16.88	
16.01/16.88	Problem 1.9: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S)
16.01/16.88	 U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S)
16.01/16.88	 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.88	 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(U21#) = 1
16.01/16.88	
16.01/16.88	Problem 1.9: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.88	 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	->->Cycle:
16.01/16.88	->->-> Pairs:
16.01/16.88	 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.88	 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.88	->->-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	
16.01/16.88	Problem 1.9: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.88	 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Projection:
16.01/16.88	 pi(U21#) = 2
16.01/16.88	
16.01/16.88	Problem 1.9: 
16.01/16.88	
16.01/16.88	SCC Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 Empty
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.88	 active(U13(tt)) -> mark(tt)
16.01/16.88	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.88	 active(U22(tt)) -> mark(tt)
16.01/16.88	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.88	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.88	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.88	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.88	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.88	 active(isNat(0)) -> mark(tt)
16.01/16.88	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.88	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.88	 active(isNatKind(0)) -> mark(tt)
16.01/16.88	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.88	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.88	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.88	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.88	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.88	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.88	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.88	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.88	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.88	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.88	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.88	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.88	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.88	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.88	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.88	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.88	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.88	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.88	 mark(0) -> active(0)
16.01/16.88	 mark(tt) -> active(tt)
16.01/16.88	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.88	 s(active(X:S)) -> s(X:S)
16.01/16.88	 s(mark(X:S)) -> s(X:S)
16.01/16.88	->Strongly Connected Components:
16.01/16.88	 There is no strongly connected component
16.01/16.88	
16.01/16.88	The problem is finite.
16.01/16.88	
16.01/16.88	Problem 1.10: 
16.01/16.88	
16.01/16.88	Subterm Processor:
16.01/16.88	-> Pairs:
16.01/16.88	 U13#(active(X:S)) -> U13#(X:S)
16.01/16.88	 U13#(mark(X:S)) -> U13#(X:S)
16.01/16.88	-> Rules:
16.01/16.88	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.88	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.88	 U13(active(X:S)) -> U13(X:S)
16.01/16.88	 U13(mark(X:S)) -> U13(X:S)
16.01/16.88	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.88	 U22(active(X:S)) -> U22(X:S)
16.01/16.88	 U22(mark(X:S)) -> U22(X:S)
16.01/16.88	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.88	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.88	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.88	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Projection:
16.01/16.89	 pi(U13#) = 1
16.01/16.89	
16.01/16.89	Problem 1.10: 
16.01/16.89	
16.01/16.89	SCC Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 Empty
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Strongly Connected Components:
16.01/16.89	 There is no strongly connected component
16.01/16.89	
16.01/16.89	The problem is finite.
16.01/16.89	
16.01/16.89	Problem 1.11: 
16.01/16.89	
16.01/16.89	Subterm Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 U12#(active(X1:S),X2:S) -> U12#(X1:S,X2:S)
16.01/16.89	 U12#(mark(X1:S),X2:S) -> U12#(X1:S,X2:S)
16.01/16.89	 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.89	 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Projection:
16.01/16.89	 pi(U12#) = 1
16.01/16.89	
16.01/16.89	Problem 1.11: 
16.01/16.89	
16.01/16.89	SCC Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.89	 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Strongly Connected Components:
16.01/16.89	->->Cycle:
16.01/16.89	->->-> Pairs:
16.01/16.89	 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.89	 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.89	->->-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	
16.01/16.89	Problem 1.11: 
16.01/16.89	
16.01/16.89	Subterm Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.89	 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Projection:
16.01/16.89	 pi(U12#) = 2
16.01/16.89	
16.01/16.89	Problem 1.11: 
16.01/16.89	
16.01/16.89	SCC Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 Empty
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Strongly Connected Components:
16.01/16.89	 There is no strongly connected component
16.01/16.89	
16.01/16.89	The problem is finite.
16.01/16.89	
16.01/16.89	Problem 1.12: 
16.01/16.89	
16.01/16.89	Subterm Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Projection:
16.01/16.89	 pi(U11#) = 1
16.01/16.89	
16.01/16.89	Problem 1.12: 
16.01/16.89	
16.01/16.89	SCC Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Strongly Connected Components:
16.01/16.89	->->Cycle:
16.01/16.89	->->-> Pairs:
16.01/16.89	 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	->->-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	
16.01/16.89	Problem 1.12: 
16.01/16.89	
16.01/16.89	Subterm Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Projection:
16.01/16.89	 pi(U11#) = 2
16.01/16.89	
16.01/16.89	Problem 1.12: 
16.01/16.89	
16.01/16.89	SCC Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Strongly Connected Components:
16.01/16.89	->->Cycle:
16.01/16.89	->->-> Pairs:
16.01/16.89	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	->->-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	
16.01/16.89	Problem 1.12: 
16.01/16.89	
16.01/16.89	Subterm Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Projection:
16.01/16.89	 pi(U11#) = 3
16.01/16.89	
16.01/16.89	Problem 1.12: 
16.01/16.89	
16.01/16.89	SCC Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 Empty
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Strongly Connected Components:
16.01/16.89	 There is no strongly connected component
16.01/16.89	
16.01/16.89	The problem is finite.
16.01/16.89	
16.01/16.89	Problem 1.13: 
16.01/16.89	
16.01/16.89	Reduction Pairs Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.89	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.89	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.89	 ACTIVE(U31(tt,N:S)) -> MARK(N:S)
16.01/16.89	 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
16.01/16.89	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.89	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.89	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.89	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.89	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.89	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.89	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.89	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.89	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	-> Usable rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Interpretation type:
16.01/16.89	 Linear
16.01/16.89	->Coefficients:
16.01/16.89	 Natural Numbers
16.01/16.89	->Dimension:
16.01/16.89	 1
16.01/16.89	->Bound:
16.01/16.89	 2
16.01/16.89	->Interpretation:
16.01/16.89	 
16.01/16.89	[U11](X1,X2,X3) = X1
16.01/16.89	[U12](X1,X2) = 2.X1
16.01/16.89	[U13](X) = X
16.01/16.89	[U21](X1,X2) = X1
16.01/16.89	[U22](X) = 2.X
16.01/16.89	[U31](X1,X2) = 2.X1 + 2.X2 + 2
16.01/16.89	[U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
16.01/16.89	[active](X) = X
16.01/16.89	[and](X1,X2) = 2.X1 + 2.X2
16.01/16.89	[isNat](X) = 0
16.01/16.89	[isNatKind](X) = 0
16.01/16.89	[mark](X) = X
16.01/16.89	[plus](X1,X2) = 2.X1 + 2.X2
16.01/16.89	[s](X) = X
16.01/16.89	[0] = 2
16.01/16.89	[fSNonEmpty] = 0
16.01/16.89	[tt] = 0
16.01/16.89	[U11#](X1,X2,X3) = 0
16.01/16.89	[U12#](X1,X2) = 0
16.01/16.89	[U13#](X) = 0
16.01/16.89	[U21#](X1,X2) = 0
16.01/16.89	[U22#](X) = 0
16.01/16.89	[U31#](X1,X2) = 0
16.01/16.89	[U41#](X1,X2,X3) = 0
16.01/16.89	[ACTIVE](X) = 2.X + 2
16.01/16.89	[AND](X1,X2) = 0
16.01/16.89	[ISNAT](X) = 0
16.01/16.89	[ISNATKIND](X) = 0
16.01/16.89	[MARK](X) = 2.X + 2
16.01/16.89	[PLUS](X1,X2) = 0
16.01/16.89	[S](X) = 0
16.01/16.89	
16.01/16.89	Problem 1.13: 
16.01/16.89	
16.01/16.89	SCC Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.89	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.89	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.89	 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
16.01/16.89	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.89	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.89	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.89	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.89	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.89	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.89	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.89	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.89	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Strongly Connected Components:
16.01/16.89	->->Cycle:
16.01/16.89	->->-> Pairs:
16.01/16.89	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.89	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.89	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.89	 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
16.01/16.89	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.89	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.89	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.89	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.89	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.89	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.89	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.89	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.89	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.89	->->-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	
16.01/16.89	Problem 1.13: 
16.01/16.89	
16.01/16.89	Reduction Pairs Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.89	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.89	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.89	 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
16.01/16.89	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.89	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.89	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.89	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.89	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.89	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.89	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.89	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.89	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	-> Usable rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Interpretation type:
16.01/16.89	 Linear
16.01/16.89	->Coefficients:
16.01/16.89	 Natural Numbers
16.01/16.89	->Dimension:
16.01/16.89	 1
16.01/16.89	->Bound:
16.01/16.89	 2
16.01/16.89	->Interpretation:
16.01/16.89	 
16.01/16.89	[U11](X1,X2,X3) = X1
16.01/16.89	[U12](X1,X2) = 2.X1
16.01/16.89	[U13](X) = 2.X
16.01/16.89	[U21](X1,X2) = 2.X1
16.01/16.89	[U22](X) = X
16.01/16.89	[U31](X1,X2) = 2.X1 + 2.X2
16.01/16.89	[U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
16.01/16.89	[active](X) = X
16.01/16.89	[and](X1,X2) = 2.X1 + 2.X2
16.01/16.89	[isNat](X) = 0
16.01/16.89	[isNatKind](X) = 0
16.01/16.89	[mark](X) = X
16.01/16.89	[plus](X1,X2) = 2.X1 + 2.X2
16.01/16.89	[s](X) = X + 1
16.01/16.89	[0] = 0
16.01/16.89	[fSNonEmpty] = 0
16.01/16.89	[tt] = 0
16.01/16.89	[U11#](X1,X2,X3) = 0
16.01/16.89	[U12#](X1,X2) = 0
16.01/16.89	[U13#](X) = 0
16.01/16.89	[U21#](X1,X2) = 0
16.01/16.89	[U22#](X) = 0
16.01/16.89	[U31#](X1,X2) = 0
16.01/16.89	[U41#](X1,X2,X3) = 0
16.01/16.89	[ACTIVE](X) = 2.X + 2
16.01/16.89	[AND](X1,X2) = 0
16.01/16.89	[ISNAT](X) = 0
16.01/16.89	[ISNATKIND](X) = 0
16.01/16.89	[MARK](X) = 2.X + 2
16.01/16.89	[PLUS](X1,X2) = 0
16.01/16.89	[S](X) = 0
16.01/16.89	
16.01/16.89	Problem 1.13: 
16.01/16.89	
16.01/16.89	SCC Processor:
16.01/16.89	-> Pairs:
16.01/16.89	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.89	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.89	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.89	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.89	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.89	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.89	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.89	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.89	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.89	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.89	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.89	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.89	-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.89	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.89	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.89	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.89	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.89	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.89	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.89	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.89	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.89	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.89	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.89	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.89	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.89	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.89	 mark(0) -> active(0)
16.01/16.89	 mark(tt) -> active(tt)
16.01/16.89	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.89	 s(active(X:S)) -> s(X:S)
16.01/16.89	 s(mark(X:S)) -> s(X:S)
16.01/16.89	->Strongly Connected Components:
16.01/16.89	->->Cycle:
16.01/16.89	->->-> Pairs:
16.01/16.89	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.89	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.89	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.89	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.89	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.89	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.89	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.89	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.89	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.89	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.89	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.89	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.89	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.89	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.89	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.89	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.89	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.89	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.89	->->-> Rules:
16.01/16.89	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.89	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.89	 U13(active(X:S)) -> U13(X:S)
16.01/16.89	 U13(mark(X:S)) -> U13(X:S)
16.01/16.89	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.89	 U22(active(X:S)) -> U22(X:S)
16.01/16.89	 U22(mark(X:S)) -> U22(X:S)
16.01/16.89	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.89	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.89	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.89	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.89	 active(U13(tt)) -> mark(tt)
16.01/16.89	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.89	 active(U22(tt)) -> mark(tt)
16.01/16.89	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.89	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.89	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.89	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.89	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.89	 active(isNat(0)) -> mark(tt)
16.01/16.89	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.89	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.89	 active(isNatKind(0)) -> mark(tt)
16.01/16.89	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.89	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.89	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.89	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.90	 mark(0) -> active(0)
16.01/16.90	 mark(tt) -> active(tt)
16.01/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 s(active(X:S)) -> s(X:S)
16.01/16.90	 s(mark(X:S)) -> s(X:S)
16.01/16.90	
16.01/16.90	Problem 1.13: 
16.01/16.90	
16.01/16.90	Reduction Pairs Processor:
16.01/16.90	-> Pairs:
16.01/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.90	 ACTIVE(plus(N:S,s(M:S))) -> MARK(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.90	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.90	-> Rules:
16.01/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U13(active(X:S)) -> U13(X:S)
16.01/16.90	 U13(mark(X:S)) -> U13(X:S)
16.01/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U22(active(X:S)) -> U22(X:S)
16.01/16.90	 U22(mark(X:S)) -> U22(X:S)
16.01/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.90	 active(U13(tt)) -> mark(tt)
16.01/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.90	 active(U22(tt)) -> mark(tt)
16.01/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 active(isNat(0)) -> mark(tt)
16.01/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.90	 active(isNatKind(0)) -> mark(tt)
16.01/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.90	 mark(0) -> active(0)
16.01/16.90	 mark(tt) -> active(tt)
16.01/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 s(active(X:S)) -> s(X:S)
16.01/16.90	 s(mark(X:S)) -> s(X:S)
16.01/16.90	-> Usable rules:
16.01/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U13(active(X:S)) -> U13(X:S)
16.01/16.90	 U13(mark(X:S)) -> U13(X:S)
16.01/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U22(active(X:S)) -> U22(X:S)
16.01/16.90	 U22(mark(X:S)) -> U22(X:S)
16.01/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.90	 active(U13(tt)) -> mark(tt)
16.01/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.90	 active(U22(tt)) -> mark(tt)
16.01/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 active(isNat(0)) -> mark(tt)
16.01/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.90	 active(isNatKind(0)) -> mark(tt)
16.01/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.90	 mark(0) -> active(0)
16.01/16.90	 mark(tt) -> active(tt)
16.01/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 s(active(X:S)) -> s(X:S)
16.01/16.90	 s(mark(X:S)) -> s(X:S)
16.01/16.90	->Interpretation type:
16.01/16.90	 Linear
16.01/16.90	->Coefficients:
16.01/16.90	 Natural Numbers
16.01/16.90	->Dimension:
16.01/16.90	 1
16.01/16.90	->Bound:
16.01/16.90	 2
16.01/16.90	->Interpretation:
16.01/16.90	 
16.01/16.90	[U11](X1,X2,X3) = 2.X1
16.01/16.90	[U12](X1,X2) = 2.X1
16.01/16.90	[U13](X) = 2.X
16.01/16.90	[U21](X1,X2) = X1
16.01/16.90	[U22](X) = X
16.01/16.90	[U31](X1,X2) = 2.X1 + 2.X2 + 1
16.01/16.90	[U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.01/16.90	[active](X) = X
16.01/16.90	[and](X1,X2) = 2.X1 + 2.X2
16.01/16.90	[isNat](X) = 0
16.01/16.90	[isNatKind](X) = 0
16.01/16.90	[mark](X) = X
16.01/16.90	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.01/16.90	[s](X) = X + 1
16.01/16.90	[0] = 0
16.01/16.90	[fSNonEmpty] = 0
16.01/16.90	[tt] = 0
16.01/16.90	[U11#](X1,X2,X3) = 0
16.01/16.90	[U12#](X1,X2) = 0
16.01/16.90	[U13#](X) = 0
16.01/16.90	[U21#](X1,X2) = 0
16.01/16.90	[U22#](X) = 0
16.01/16.90	[U31#](X1,X2) = 0
16.01/16.90	[U41#](X1,X2,X3) = 0
16.01/16.90	[ACTIVE](X) = 2.X + 2
16.01/16.90	[AND](X1,X2) = 0
16.01/16.90	[ISNAT](X) = 0
16.01/16.90	[ISNATKIND](X) = 0
16.01/16.90	[MARK](X) = 2.X + 2
16.01/16.90	[PLUS](X1,X2) = 0
16.01/16.90	[S](X) = 0
16.01/16.90	
16.01/16.90	Problem 1.13: 
16.01/16.90	
16.01/16.90	SCC Processor:
16.01/16.90	-> Pairs:
16.01/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.90	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.90	-> Rules:
16.01/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U13(active(X:S)) -> U13(X:S)
16.01/16.90	 U13(mark(X:S)) -> U13(X:S)
16.01/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U22(active(X:S)) -> U22(X:S)
16.01/16.90	 U22(mark(X:S)) -> U22(X:S)
16.01/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.90	 active(U13(tt)) -> mark(tt)
16.01/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.90	 active(U22(tt)) -> mark(tt)
16.01/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 active(isNat(0)) -> mark(tt)
16.01/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.90	 active(isNatKind(0)) -> mark(tt)
16.01/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.90	 mark(0) -> active(0)
16.01/16.90	 mark(tt) -> active(tt)
16.01/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 s(active(X:S)) -> s(X:S)
16.01/16.90	 s(mark(X:S)) -> s(X:S)
16.01/16.90	->Strongly Connected Components:
16.01/16.90	->->Cycle:
16.01/16.90	->->-> Pairs:
16.01/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.90	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.90	->->-> Rules:
16.01/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U13(active(X:S)) -> U13(X:S)
16.01/16.90	 U13(mark(X:S)) -> U13(X:S)
16.01/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U22(active(X:S)) -> U22(X:S)
16.01/16.90	 U22(mark(X:S)) -> U22(X:S)
16.01/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.90	 active(U13(tt)) -> mark(tt)
16.01/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.90	 active(U22(tt)) -> mark(tt)
16.01/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 active(isNat(0)) -> mark(tt)
16.01/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.90	 active(isNatKind(0)) -> mark(tt)
16.01/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.90	 mark(0) -> active(0)
16.01/16.90	 mark(tt) -> active(tt)
16.01/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 s(active(X:S)) -> s(X:S)
16.01/16.90	 s(mark(X:S)) -> s(X:S)
16.01/16.90	
16.01/16.90	Problem 1.13: 
16.01/16.90	
16.01/16.90	Reduction Pairs Processor:
16.01/16.90	-> Pairs:
16.01/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.90	 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.01/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.01/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.01/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.01/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.01/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.01/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.01/16.90	-> Rules:
16.01/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U13(active(X:S)) -> U13(X:S)
16.01/16.90	 U13(mark(X:S)) -> U13(X:S)
16.01/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U22(active(X:S)) -> U22(X:S)
16.01/16.90	 U22(mark(X:S)) -> U22(X:S)
16.01/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.90	 active(U13(tt)) -> mark(tt)
16.01/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.90	 active(U22(tt)) -> mark(tt)
16.01/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 active(isNat(0)) -> mark(tt)
16.01/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.90	 active(isNatKind(0)) -> mark(tt)
16.01/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.90	 mark(0) -> active(0)
16.01/16.90	 mark(tt) -> active(tt)
16.01/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 s(active(X:S)) -> s(X:S)
16.01/16.90	 s(mark(X:S)) -> s(X:S)
16.01/16.90	-> Usable rules:
16.01/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.01/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.01/16.90	 U13(active(X:S)) -> U13(X:S)
16.01/16.90	 U13(mark(X:S)) -> U13(X:S)
16.01/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.01/16.90	 U22(active(X:S)) -> U22(X:S)
16.01/16.90	 U22(mark(X:S)) -> U22(X:S)
16.01/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.01/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.01/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.01/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.01/16.90	 active(U13(tt)) -> mark(tt)
16.01/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.01/16.90	 active(U22(tt)) -> mark(tt)
16.01/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.01/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.01/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.01/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 active(isNat(0)) -> mark(tt)
16.01/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.01/16.90	 active(isNatKind(0)) -> mark(tt)
16.01/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.01/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.01/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.01/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.01/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.01/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.01/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.01/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.01/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.01/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.01/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.01/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.01/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.01/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.01/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.01/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.01/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.01/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.01/16.90	 mark(0) -> active(0)
16.01/16.90	 mark(tt) -> active(tt)
16.01/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.01/16.90	 s(active(X:S)) -> s(X:S)
16.01/16.90	 s(mark(X:S)) -> s(X:S)
16.01/16.90	->Interpretation type:
16.01/16.90	 Linear
16.01/16.90	->Coefficients:
16.01/16.90	 Natural Numbers
16.01/16.90	->Dimension:
16.01/16.90	 1
16.01/16.90	->Bound:
16.01/16.90	 2
16.01/16.90	->Interpretation:
16.01/16.90	 
16.01/16.90	[U11](X1,X2,X3) = 2.X1
16.01/16.90	[U12](X1,X2) = 2.X1
16.01/16.90	[U13](X) = 2.X
16.01/16.90	[U21](X1,X2) = 2.X1
16.01/16.90	[U22](X) = X
16.01/16.90	[U31](X1,X2) = 2.X1 + 2.X2 + 1
16.01/16.90	[U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.01/16.90	[active](X) = X
16.01/16.90	[and](X1,X2) = 2.X1 + X2
16.01/16.90	[isNat](X) = 0
16.01/16.90	[isNatKind](X) = 0
16.01/16.90	[mark](X) = X
16.01/16.90	[plus](X1,X2) = 2.X1 + 2.X2 + 2
16.01/16.90	[s](X) = X
16.01/16.90	[0] = 1
16.01/16.90	[fSNonEmpty] = 0
16.01/16.90	[tt] = 0
16.01/16.90	[U11#](X1,X2,X3) = 0
16.01/16.90	[U12#](X1,X2) = 0
16.01/16.90	[U13#](X) = 0
16.01/16.90	[U21#](X1,X2) = 0
16.01/16.90	[U22#](X) = 0
16.01/16.90	[U31#](X1,X2) = 0
16.01/16.90	[U41#](X1,X2,X3) = 0
16.01/16.90	[ACTIVE](X) = 2.X + 2
16.01/16.90	[AND](X1,X2) = 0
16.01/16.90	[ISNAT](X) = 0
16.01/16.90	[ISNATKIND](X) = 0
16.01/16.90	[MARK](X) = 2.X + 2
16.01/16.90	[PLUS](X1,X2) = 0
16.01/16.90	[S](X) = 0
16.01/16.90	
16.01/16.90	Problem 1.13: 
16.01/16.90	
16.01/16.90	SCC Processor:
16.01/16.90	-> Pairs:
16.01/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.01/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.01/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.01/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.01/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.01/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.01/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.01/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.01/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.01/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.01/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.01/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.01/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.01/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	->Strongly Connected Components:
16.04/16.90	->->Cycle:
16.04/16.90	->->-> Pairs:
16.04/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.04/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.04/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	->->-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	
16.04/16.90	Problem 1.13: 
16.04/16.90	
16.04/16.90	Reduction Pairs Processor:
16.04/16.90	-> Pairs:
16.04/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
16.04/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.04/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	-> Usable rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	->Interpretation type:
16.04/16.90	 Linear
16.04/16.90	->Coefficients:
16.04/16.90	 Natural Numbers
16.04/16.90	->Dimension:
16.04/16.90	 1
16.04/16.90	->Bound:
16.04/16.90	 2
16.04/16.90	->Interpretation:
16.04/16.90	 
16.04/16.90	[U11](X1,X2,X3) = 1
16.04/16.90	[U12](X1,X2) = 1
16.04/16.90	[U13](X) = 0
16.04/16.90	[U21](X1,X2) = 1
16.04/16.90	[U22](X) = 1
16.04/16.90	[U31](X1,X2) = 1
16.04/16.90	[U41](X1,X2,X3) = 1
16.04/16.90	[active](X) = 2
16.04/16.90	[and](X1,X2) = 1
16.04/16.90	[isNat](X) = 1
16.04/16.90	[isNatKind](X) = 1
16.04/16.90	[mark](X) = 2
16.04/16.90	[plus](X1,X2) = 0
16.04/16.90	[s](X) = 1
16.04/16.90	[0] = 1
16.04/16.90	[fSNonEmpty] = 0
16.04/16.90	[tt] = 0
16.04/16.90	[U11#](X1,X2,X3) = 0
16.04/16.90	[U12#](X1,X2) = 0
16.04/16.90	[U13#](X) = 0
16.04/16.90	[U21#](X1,X2) = 0
16.04/16.90	[U22#](X) = 0
16.04/16.90	[U31#](X1,X2) = 0
16.04/16.90	[U41#](X1,X2,X3) = 0
16.04/16.90	[ACTIVE](X) = 2.X
16.04/16.90	[AND](X1,X2) = 0
16.04/16.90	[ISNAT](X) = 0
16.04/16.90	[ISNATKIND](X) = 0
16.04/16.90	[MARK](X) = 2
16.04/16.90	[PLUS](X1,X2) = 0
16.04/16.90	[S](X) = 0
16.04/16.90	
16.04/16.90	Problem 1.13: 
16.04/16.90	
16.04/16.90	SCC Processor:
16.04/16.90	-> Pairs:
16.04/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.04/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	->Strongly Connected Components:
16.04/16.90	->->Cycle:
16.04/16.90	->->-> Pairs:
16.04/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.04/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	->->-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	
16.04/16.90	Problem 1.13: 
16.04/16.90	
16.04/16.90	Reduction Pairs Processor:
16.04/16.90	-> Pairs:
16.04/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
16.04/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	-> Usable rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	->Interpretation type:
16.04/16.90	 Linear
16.04/16.90	->Coefficients:
16.04/16.90	 Natural Numbers
16.04/16.90	->Dimension:
16.04/16.90	 1
16.04/16.90	->Bound:
16.04/16.90	 2
16.04/16.90	->Interpretation:
16.04/16.90	 
16.04/16.90	[U11](X1,X2,X3) = 1
16.04/16.90	[U12](X1,X2) = 1
16.04/16.90	[U13](X) = 2
16.04/16.90	[U21](X1,X2) = 1
16.04/16.90	[U22](X) = 0
16.04/16.90	[U31](X1,X2) = 1
16.04/16.90	[U41](X1,X2,X3) = 1
16.04/16.90	[active](X) = 2
16.04/16.90	[and](X1,X2) = 1
16.04/16.90	[isNat](X) = 1
16.04/16.90	[isNatKind](X) = 1
16.04/16.90	[mark](X) = 2
16.04/16.90	[plus](X1,X2) = 0
16.04/16.90	[s](X) = 1
16.04/16.90	[0] = 2
16.04/16.90	[fSNonEmpty] = 0
16.04/16.90	[tt] = 2
16.04/16.90	[U11#](X1,X2,X3) = 0
16.04/16.90	[U12#](X1,X2) = 0
16.04/16.90	[U13#](X) = 0
16.04/16.90	[U21#](X1,X2) = 0
16.04/16.90	[U22#](X) = 0
16.04/16.90	[U31#](X1,X2) = 0
16.04/16.90	[U41#](X1,X2,X3) = 0
16.04/16.90	[ACTIVE](X) = 2.X
16.04/16.90	[AND](X1,X2) = 0
16.04/16.90	[ISNAT](X) = 0
16.04/16.90	[ISNATKIND](X) = 0
16.04/16.90	[MARK](X) = 2
16.04/16.90	[PLUS](X1,X2) = 0
16.04/16.90	[S](X) = 0
16.04/16.90	
16.04/16.90	Problem 1.13: 
16.04/16.90	
16.04/16.90	SCC Processor:
16.04/16.90	-> Pairs:
16.04/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	->Strongly Connected Components:
16.04/16.90	->->Cycle:
16.04/16.90	->->-> Pairs:
16.04/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	->->-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	
16.04/16.90	Problem 1.13: 
16.04/16.90	
16.04/16.90	Reduction Pairs Processor:
16.04/16.90	-> Pairs:
16.04/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	-> Usable rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.90	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.90	 active(U13(tt)) -> mark(tt)
16.04/16.90	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.90	 active(U22(tt)) -> mark(tt)
16.04/16.90	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.90	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.90	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.90	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 active(isNat(0)) -> mark(tt)
16.04/16.90	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.90	 active(isNatKind(0)) -> mark(tt)
16.04/16.90	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.90	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.90	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.90	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.90	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.90	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.90	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.90	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.90	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.90	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.90	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.90	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.90	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.90	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.90	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.90	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.90	 mark(0) -> active(0)
16.04/16.90	 mark(tt) -> active(tt)
16.04/16.90	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.90	 s(active(X:S)) -> s(X:S)
16.04/16.90	 s(mark(X:S)) -> s(X:S)
16.04/16.90	->Interpretation type:
16.04/16.90	 Linear
16.04/16.90	->Coefficients:
16.04/16.90	 Natural Numbers
16.04/16.90	->Dimension:
16.04/16.90	 1
16.04/16.90	->Bound:
16.04/16.90	 2
16.04/16.90	->Interpretation:
16.04/16.90	 
16.04/16.90	[U11](X1,X2,X3) = 1
16.04/16.90	[U12](X1,X2) = 1
16.04/16.90	[U13](X) = 2
16.04/16.90	[U21](X1,X2) = 1
16.04/16.90	[U22](X) = 2
16.04/16.90	[U31](X1,X2) = 0
16.04/16.90	[U41](X1,X2,X3) = 0
16.04/16.90	[active](X) = 2
16.04/16.90	[and](X1,X2) = 1
16.04/16.90	[isNat](X) = 1
16.04/16.90	[isNatKind](X) = 1
16.04/16.90	[mark](X) = 2
16.04/16.90	[plus](X1,X2) = 1
16.04/16.90	[s](X) = 1
16.04/16.90	[0] = 0
16.04/16.90	[fSNonEmpty] = 0
16.04/16.90	[tt] = 2
16.04/16.90	[U11#](X1,X2,X3) = 0
16.04/16.90	[U12#](X1,X2) = 0
16.04/16.90	[U13#](X) = 0
16.04/16.90	[U21#](X1,X2) = 0
16.04/16.90	[U22#](X) = 0
16.04/16.90	[U31#](X1,X2) = 0
16.04/16.90	[U41#](X1,X2,X3) = 0
16.04/16.90	[ACTIVE](X) = 2.X
16.04/16.90	[AND](X1,X2) = 0
16.04/16.90	[ISNAT](X) = 0
16.04/16.90	[ISNATKIND](X) = 0
16.04/16.90	[MARK](X) = 2
16.04/16.90	[PLUS](X1,X2) = 0
16.04/16.90	[S](X) = 0
16.04/16.90	
16.04/16.90	Problem 1.13: 
16.04/16.90	
16.04/16.90	SCC Processor:
16.04/16.90	-> Pairs:
16.04/16.90	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.90	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.90	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.90	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.90	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.90	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.90	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.90	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.90	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.90	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.90	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.90	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.90	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.90	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.90	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.90	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.90	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.90	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.90	-> Rules:
16.04/16.90	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.90	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.90	 U13(active(X:S)) -> U13(X:S)
16.04/16.90	 U13(mark(X:S)) -> U13(X:S)
16.04/16.90	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.90	 U22(active(X:S)) -> U22(X:S)
16.04/16.90	 U22(mark(X:S)) -> U22(X:S)
16.04/16.90	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.90	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.90	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Strongly Connected Components:
16.04/16.91	->->Cycle:
16.04/16.91	->->-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	->->-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	Reduction Pairs Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	-> Usable rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Interpretation type:
16.04/16.91	 Linear
16.04/16.91	->Coefficients:
16.04/16.91	 Natural Numbers
16.04/16.91	->Dimension:
16.04/16.91	 1
16.04/16.91	->Bound:
16.04/16.91	 2
16.04/16.91	->Interpretation:
16.04/16.91	 
16.04/16.91	[U11](X1,X2,X3) = 2.X1
16.04/16.91	[U12](X1,X2) = 2.X1
16.04/16.91	[U13](X) = 2.X
16.04/16.91	[U21](X1,X2) = 2.X1
16.04/16.91	[U22](X) = 2.X
16.04/16.91	[U31](X1,X2) = X1 + X2 + 1
16.04/16.91	[U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.04/16.91	[active](X) = X
16.04/16.91	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.91	[isNat](X) = 0
16.04/16.91	[isNatKind](X) = 0
16.04/16.91	[mark](X) = X
16.04/16.91	[plus](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.91	[s](X) = X
16.04/16.91	[0] = 2
16.04/16.91	[fSNonEmpty] = 0
16.04/16.91	[tt] = 0
16.04/16.91	[U11#](X1,X2,X3) = 0
16.04/16.91	[U12#](X1,X2) = 0
16.04/16.91	[U13#](X) = 0
16.04/16.91	[U21#](X1,X2) = 0
16.04/16.91	[U22#](X) = 0
16.04/16.91	[U31#](X1,X2) = 0
16.04/16.91	[U41#](X1,X2,X3) = 0
16.04/16.91	[ACTIVE](X) = 2.X + 1
16.04/16.91	[AND](X1,X2) = 0
16.04/16.91	[ISNAT](X) = 0
16.04/16.91	[ISNATKIND](X) = 0
16.04/16.91	[MARK](X) = 2.X + 1
16.04/16.91	[PLUS](X1,X2) = 0
16.04/16.91	[S](X) = 0
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	SCC Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Strongly Connected Components:
16.04/16.91	->->Cycle:
16.04/16.91	->->-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	->->-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	Reduction Pairs Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	-> Usable rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Interpretation type:
16.04/16.91	 Linear
16.04/16.91	->Coefficients:
16.04/16.91	 Natural Numbers
16.04/16.91	->Dimension:
16.04/16.91	 1
16.04/16.91	->Bound:
16.04/16.91	 2
16.04/16.91	->Interpretation:
16.04/16.91	 
16.04/16.91	[U11](X1,X2,X3) = X1
16.04/16.91	[U12](X1,X2) = 2.X1
16.04/16.91	[U13](X) = 2.X
16.04/16.91	[U21](X1,X2) = X1
16.04/16.91	[U22](X) = 2.X
16.04/16.91	[U31](X1,X2) = X1 + 2.X2 + 2
16.04/16.91	[U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
16.04/16.91	[active](X) = X
16.04/16.91	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.91	[isNat](X) = 0
16.04/16.91	[isNatKind](X) = 0
16.04/16.91	[mark](X) = X
16.04/16.91	[plus](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.91	[s](X) = X
16.04/16.91	[0] = 2
16.04/16.91	[fSNonEmpty] = 0
16.04/16.91	[tt] = 0
16.04/16.91	[U11#](X1,X2,X3) = 0
16.04/16.91	[U12#](X1,X2) = 0
16.04/16.91	[U13#](X) = 0
16.04/16.91	[U21#](X1,X2) = 0
16.04/16.91	[U22#](X) = 0
16.04/16.91	[U31#](X1,X2) = 0
16.04/16.91	[U41#](X1,X2,X3) = 0
16.04/16.91	[ACTIVE](X) = X + 2
16.04/16.91	[AND](X1,X2) = 0
16.04/16.91	[ISNAT](X) = 0
16.04/16.91	[ISNATKIND](X) = 0
16.04/16.91	[MARK](X) = 2.X + 2
16.04/16.91	[PLUS](X1,X2) = 0
16.04/16.91	[S](X) = 0
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	SCC Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Strongly Connected Components:
16.04/16.91	->->Cycle:
16.04/16.91	->->-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	->->-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	Reduction Pairs Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	-> Usable rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Interpretation type:
16.04/16.91	 Linear
16.04/16.91	->Coefficients:
16.04/16.91	 Natural Numbers
16.04/16.91	->Dimension:
16.04/16.91	 1
16.04/16.91	->Bound:
16.04/16.91	 2
16.04/16.91	->Interpretation:
16.04/16.91	 
16.04/16.91	[U11](X1,X2,X3) = 2.X1
16.04/16.91	[U12](X1,X2) = X1
16.04/16.91	[U13](X) = X
16.04/16.91	[U21](X1,X2) = X1
16.04/16.91	[U22](X) = X
16.04/16.91	[U31](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.91	[U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
16.04/16.91	[active](X) = X
16.04/16.91	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.91	[isNat](X) = 0
16.04/16.91	[isNatKind](X) = 0
16.04/16.91	[mark](X) = X
16.04/16.91	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.91	[s](X) = X + 1
16.04/16.91	[0] = 2
16.04/16.91	[fSNonEmpty] = 0
16.04/16.91	[tt] = 0
16.04/16.91	[U11#](X1,X2,X3) = 0
16.04/16.91	[U12#](X1,X2) = 0
16.04/16.91	[U13#](X) = 0
16.04/16.91	[U21#](X1,X2) = 0
16.04/16.91	[U22#](X) = 0
16.04/16.91	[U31#](X1,X2) = 0
16.04/16.91	[U41#](X1,X2,X3) = 0
16.04/16.91	[ACTIVE](X) = 2.X + 2
16.04/16.91	[AND](X1,X2) = 0
16.04/16.91	[ISNAT](X) = 0
16.04/16.91	[ISNATKIND](X) = 0
16.04/16.91	[MARK](X) = 2.X + 2
16.04/16.91	[PLUS](X1,X2) = 0
16.04/16.91	[S](X) = 0
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	SCC Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Strongly Connected Components:
16.04/16.91	->->Cycle:
16.04/16.91	->->-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	->->-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	Reduction Pairs Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	-> Usable rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Interpretation type:
16.04/16.91	 Linear
16.04/16.91	->Coefficients:
16.04/16.91	 Natural Numbers
16.04/16.91	->Dimension:
16.04/16.91	 1
16.04/16.91	->Bound:
16.04/16.91	 2
16.04/16.91	->Interpretation:
16.04/16.91	 
16.04/16.91	[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.04/16.91	[U12](X1,X2) = X1 + 2.X2 + 1
16.04/16.91	[U13](X) = X + 1
16.04/16.91	[U21](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.91	[U22](X) = X + 2
16.04/16.91	[U31](X1,X2) = 2.X2 + 2
16.04/16.91	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.91	[active](X) = X
16.04/16.91	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.91	[isNat](X) = 2.X
16.04/16.91	[isNatKind](X) = 0
16.04/16.91	[mark](X) = X
16.04/16.91	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.91	[s](X) = X + 1
16.04/16.91	[0] = 2
16.04/16.91	[fSNonEmpty] = 0
16.04/16.91	[tt] = 0
16.04/16.91	[U11#](X1,X2,X3) = 0
16.04/16.91	[U12#](X1,X2) = 0
16.04/16.91	[U13#](X) = 0
16.04/16.91	[U21#](X1,X2) = 0
16.04/16.91	[U22#](X) = 0
16.04/16.91	[U31#](X1,X2) = 0
16.04/16.91	[U41#](X1,X2,X3) = 0
16.04/16.91	[ACTIVE](X) = 2.X + 2
16.04/16.91	[AND](X1,X2) = 0
16.04/16.91	[ISNAT](X) = 0
16.04/16.91	[ISNATKIND](X) = 0
16.04/16.91	[MARK](X) = 2.X + 2
16.04/16.91	[PLUS](X1,X2) = 0
16.04/16.91	[S](X) = 0
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	SCC Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Strongly Connected Components:
16.04/16.91	->->Cycle:
16.04/16.91	->->-> Pairs:
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	->->-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	Reduction Pairs Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.91	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.91	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.91	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.91	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.91	-> Rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	-> Usable rules:
16.04/16.91	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.91	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.91	 U13(active(X:S)) -> U13(X:S)
16.04/16.91	 U13(mark(X:S)) -> U13(X:S)
16.04/16.91	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.91	 U22(active(X:S)) -> U22(X:S)
16.04/16.91	 U22(mark(X:S)) -> U22(X:S)
16.04/16.91	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.91	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.91	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.91	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.91	 active(U13(tt)) -> mark(tt)
16.04/16.91	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.91	 active(U22(tt)) -> mark(tt)
16.04/16.91	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.91	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.91	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.91	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 active(isNat(0)) -> mark(tt)
16.04/16.91	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.91	 active(isNatKind(0)) -> mark(tt)
16.04/16.91	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.91	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.91	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.91	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.91	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.91	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.91	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.91	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.91	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.91	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.91	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.91	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.91	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.91	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.91	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.91	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.91	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.91	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.91	 mark(0) -> active(0)
16.04/16.91	 mark(tt) -> active(tt)
16.04/16.91	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.91	 s(active(X:S)) -> s(X:S)
16.04/16.91	 s(mark(X:S)) -> s(X:S)
16.04/16.91	->Interpretation type:
16.04/16.91	 Linear
16.04/16.91	->Coefficients:
16.04/16.91	 Natural Numbers
16.04/16.91	->Dimension:
16.04/16.91	 1
16.04/16.91	->Bound:
16.04/16.91	 2
16.04/16.91	->Interpretation:
16.04/16.91	 
16.04/16.91	[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.04/16.91	[U12](X1,X2) = X1 + 2.X2 + 2
16.04/16.91	[U13](X) = X + 1
16.04/16.91	[U21](X1,X2) = X1 + 2.X2 + 2
16.04/16.91	[U22](X) = X + 2
16.04/16.91	[U31](X1,X2) = 2.X2 + 2
16.04/16.91	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.91	[active](X) = X
16.04/16.91	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.91	[isNat](X) = 2.X
16.04/16.91	[isNatKind](X) = 0
16.04/16.91	[mark](X) = X
16.04/16.91	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.91	[s](X) = X + 1
16.04/16.91	[0] = 1
16.04/16.91	[fSNonEmpty] = 0
16.04/16.91	[tt] = 0
16.04/16.91	[U11#](X1,X2,X3) = 0
16.04/16.91	[U12#](X1,X2) = 0
16.04/16.91	[U13#](X) = 0
16.04/16.91	[U21#](X1,X2) = 0
16.04/16.91	[U22#](X) = 0
16.04/16.91	[U31#](X1,X2) = 0
16.04/16.91	[U41#](X1,X2,X3) = 0
16.04/16.91	[ACTIVE](X) = 2.X + 2
16.04/16.91	[AND](X1,X2) = 0
16.04/16.91	[ISNAT](X) = 0
16.04/16.91	[ISNATKIND](X) = 0
16.04/16.91	[MARK](X) = 2.X + 2
16.04/16.91	[PLUS](X1,X2) = 0
16.04/16.91	[S](X) = 0
16.04/16.91	
16.04/16.91	Problem 1.13: 
16.04/16.91	
16.04/16.91	SCC Processor:
16.04/16.91	-> Pairs:
16.04/16.91	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.91	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.91	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.91	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.91	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.91	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.91	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.91	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.91	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.91	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Strongly Connected Components:
16.04/16.92	->->Cycle:
16.04/16.92	->->-> Pairs:
16.04/16.92	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	->->-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	Reduction Pairs Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	-> Usable rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Interpretation type:
16.04/16.92	 Linear
16.04/16.92	->Coefficients:
16.04/16.92	 Natural Numbers
16.04/16.92	->Dimension:
16.04/16.92	 1
16.04/16.92	->Bound:
16.04/16.92	 2
16.04/16.92	->Interpretation:
16.04/16.92	 
16.04/16.92	[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.04/16.92	[U12](X1,X2) = X1 + 2.X2 + 2
16.04/16.92	[U13](X) = X + 2
16.04/16.92	[U21](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.92	[U22](X) = X + 1
16.04/16.92	[U31](X1,X2) = X2 + 2
16.04/16.92	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.92	[active](X) = X
16.04/16.92	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.92	[isNat](X) = 2.X
16.04/16.92	[isNatKind](X) = 0
16.04/16.92	[mark](X) = X
16.04/16.92	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.92	[s](X) = X + 1
16.04/16.92	[0] = 2
16.04/16.92	[fSNonEmpty] = 0
16.04/16.92	[tt] = 0
16.04/16.92	[U11#](X1,X2,X3) = 0
16.04/16.92	[U12#](X1,X2) = 0
16.04/16.92	[U13#](X) = 0
16.04/16.92	[U21#](X1,X2) = 0
16.04/16.92	[U22#](X) = 0
16.04/16.92	[U31#](X1,X2) = 0
16.04/16.92	[U41#](X1,X2,X3) = 0
16.04/16.92	[ACTIVE](X) = 2.X + 2
16.04/16.92	[AND](X1,X2) = 0
16.04/16.92	[ISNAT](X) = 0
16.04/16.92	[ISNATKIND](X) = 0
16.04/16.92	[MARK](X) = 2.X + 2
16.04/16.92	[PLUS](X1,X2) = 0
16.04/16.92	[S](X) = 0
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	SCC Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Strongly Connected Components:
16.04/16.92	->->Cycle:
16.04/16.92	->->-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	->->-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	Reduction Pairs Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	-> Usable rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Interpretation type:
16.04/16.92	 Linear
16.04/16.92	->Coefficients:
16.04/16.92	 Natural Numbers
16.04/16.92	->Dimension:
16.04/16.92	 1
16.04/16.92	->Bound:
16.04/16.92	 2
16.04/16.92	->Interpretation:
16.04/16.92	 
16.04/16.92	[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.04/16.92	[U12](X1,X2) = X1 + 2.X2 + 1
16.04/16.92	[U13](X) = X
16.04/16.92	[U21](X1,X2) = X1 + 2.X2 + 2
16.04/16.92	[U22](X) = X + 1
16.04/16.92	[U31](X1,X2) = 2.X2 + 2
16.04/16.92	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.92	[active](X) = X
16.04/16.92	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.92	[isNat](X) = 2.X + 1
16.04/16.92	[isNatKind](X) = 0
16.04/16.92	[mark](X) = X
16.04/16.92	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.92	[s](X) = X + 1
16.04/16.92	[0] = 1
16.04/16.92	[fSNonEmpty] = 0
16.04/16.92	[tt] = 0
16.04/16.92	[U11#](X1,X2,X3) = 0
16.04/16.92	[U12#](X1,X2) = 0
16.04/16.92	[U13#](X) = 0
16.04/16.92	[U21#](X1,X2) = 0
16.04/16.92	[U22#](X) = 0
16.04/16.92	[U31#](X1,X2) = 0
16.04/16.92	[U41#](X1,X2,X3) = 0
16.04/16.92	[ACTIVE](X) = 2.X
16.04/16.92	[AND](X1,X2) = 0
16.04/16.92	[ISNAT](X) = 0
16.04/16.92	[ISNATKIND](X) = 0
16.04/16.92	[MARK](X) = 2.X
16.04/16.92	[PLUS](X1,X2) = 0
16.04/16.92	[S](X) = 0
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	SCC Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Strongly Connected Components:
16.04/16.92	->->Cycle:
16.04/16.92	->->-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	->->-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	Reduction Pairs Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	-> Usable rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Interpretation type:
16.04/16.92	 Linear
16.04/16.92	->Coefficients:
16.04/16.92	 Natural Numbers
16.04/16.92	->Dimension:
16.04/16.92	 1
16.04/16.92	->Bound:
16.04/16.92	 2
16.04/16.92	->Interpretation:
16.04/16.92	 
16.04/16.92	[U11](X1,X2,X3) = X1 + 2.X2 + 2.X3
16.04/16.92	[U12](X1,X2) = X1 + 2.X2
16.04/16.92	[U13](X) = X
16.04/16.92	[U21](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.92	[U22](X) = X + 1
16.04/16.92	[U31](X1,X2) = 2.X2 + 1
16.04/16.92	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.92	[active](X) = X
16.04/16.92	[and](X1,X2) = X1 + 2.X2
16.04/16.92	[isNat](X) = 2.X
16.04/16.92	[isNatKind](X) = 0
16.04/16.92	[mark](X) = X
16.04/16.92	[plus](X1,X2) = 2.X1 + 2.X2
16.04/16.92	[s](X) = X + 2
16.04/16.92	[0] = 2
16.04/16.92	[fSNonEmpty] = 0
16.04/16.92	[tt] = 0
16.04/16.92	[U11#](X1,X2,X3) = 0
16.04/16.92	[U12#](X1,X2) = 0
16.04/16.92	[U13#](X) = 0
16.04/16.92	[U21#](X1,X2) = 0
16.04/16.92	[U22#](X) = 0
16.04/16.92	[U31#](X1,X2) = 0
16.04/16.92	[U41#](X1,X2,X3) = 0
16.04/16.92	[ACTIVE](X) = 2.X + 2
16.04/16.92	[AND](X1,X2) = 0
16.04/16.92	[ISNAT](X) = 0
16.04/16.92	[ISNATKIND](X) = 0
16.04/16.92	[MARK](X) = 2.X + 2
16.04/16.92	[PLUS](X1,X2) = 0
16.04/16.92	[S](X) = 0
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	SCC Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Strongly Connected Components:
16.04/16.92	->->Cycle:
16.04/16.92	->->-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	->->-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	Reduction Pairs Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	-> Usable rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Interpretation type:
16.04/16.92	 Linear
16.04/16.92	->Coefficients:
16.04/16.92	 Natural Numbers
16.04/16.92	->Dimension:
16.04/16.92	 1
16.04/16.92	->Bound:
16.04/16.92	 2
16.04/16.92	->Interpretation:
16.04/16.92	 
16.04/16.92	[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.04/16.92	[U12](X1,X2) = X1 + 2.X2 + 1
16.04/16.92	[U13](X) = X
16.04/16.92	[U21](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.92	[U22](X) = X
16.04/16.92	[U31](X1,X2) = X2 + 2
16.04/16.92	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.92	[active](X) = X
16.04/16.92	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.92	[isNat](X) = 2.X + 1
16.04/16.92	[isNatKind](X) = 0
16.04/16.92	[mark](X) = X
16.04/16.92	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.92	[s](X) = X + 1
16.04/16.92	[0] = 2
16.04/16.92	[fSNonEmpty] = 0
16.04/16.92	[tt] = 0
16.04/16.92	[U11#](X1,X2,X3) = 0
16.04/16.92	[U12#](X1,X2) = 0
16.04/16.92	[U13#](X) = 0
16.04/16.92	[U21#](X1,X2) = 0
16.04/16.92	[U22#](X) = 0
16.04/16.92	[U31#](X1,X2) = 0
16.04/16.92	[U41#](X1,X2,X3) = 0
16.04/16.92	[ACTIVE](X) = X + 2
16.04/16.92	[AND](X1,X2) = 0
16.04/16.92	[ISNAT](X) = 0
16.04/16.92	[ISNATKIND](X) = 0
16.04/16.92	[MARK](X) = 2.X + 2
16.04/16.92	[PLUS](X1,X2) = 0
16.04/16.92	[S](X) = 0
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	SCC Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Strongly Connected Components:
16.04/16.92	->->Cycle:
16.04/16.92	->->-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	->->-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	Reduction Pairs Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	-> Usable rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U13(active(X:S)) -> U13(X:S)
16.04/16.92	 U13(mark(X:S)) -> U13(X:S)
16.04/16.92	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.92	 U22(active(X:S)) -> U22(X:S)
16.04/16.92	 U22(mark(X:S)) -> U22(X:S)
16.04/16.92	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.92	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.92	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.92	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.92	 active(U13(tt)) -> mark(tt)
16.04/16.92	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.92	 active(U22(tt)) -> mark(tt)
16.04/16.92	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.92	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.92	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.92	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.92	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.92	 active(isNat(0)) -> mark(tt)
16.04/16.92	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.92	 active(isNatKind(0)) -> mark(tt)
16.04/16.92	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.92	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.92	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.92	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.92	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.92	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.92	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.92	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.92	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.92	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.92	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.92	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.92	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.92	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.92	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.92	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.92	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.92	 mark(0) -> active(0)
16.04/16.92	 mark(tt) -> active(tt)
16.04/16.92	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.92	 s(active(X:S)) -> s(X:S)
16.04/16.92	 s(mark(X:S)) -> s(X:S)
16.04/16.92	->Interpretation type:
16.04/16.92	 Linear
16.04/16.92	->Coefficients:
16.04/16.92	 Natural Numbers
16.04/16.92	->Dimension:
16.04/16.92	 1
16.04/16.92	->Bound:
16.04/16.92	 2
16.04/16.92	->Interpretation:
16.04/16.92	 
16.04/16.92	[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.04/16.92	[U12](X1,X2) = X1 + 2.X2 + 1
16.04/16.92	[U13](X) = X
16.04/16.92	[U21](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.92	[U22](X) = X
16.04/16.92	[U31](X1,X2) = 2.X2 + 2
16.04/16.92	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.92	[active](X) = X
16.04/16.92	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.92	[isNat](X) = 2.X + 1
16.04/16.92	[isNatKind](X) = 0
16.04/16.92	[mark](X) = X
16.04/16.92	[plus](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.92	[s](X) = X
16.04/16.92	[0] = 2
16.04/16.92	[fSNonEmpty] = 0
16.04/16.92	[tt] = 0
16.04/16.92	[U11#](X1,X2,X3) = 0
16.04/16.92	[U12#](X1,X2) = 0
16.04/16.92	[U13#](X) = 0
16.04/16.92	[U21#](X1,X2) = 0
16.04/16.92	[U22#](X) = 0
16.04/16.92	[U31#](X1,X2) = 0
16.04/16.92	[U41#](X1,X2,X3) = 0
16.04/16.92	[ACTIVE](X) = X + 2
16.04/16.92	[AND](X1,X2) = 0
16.04/16.92	[ISNAT](X) = 0
16.04/16.92	[ISNATKIND](X) = 0
16.04/16.92	[MARK](X) = 2.X + 2
16.04/16.92	[PLUS](X1,X2) = 0
16.04/16.92	[S](X) = 0
16.04/16.92	
16.04/16.92	Problem 1.13: 
16.04/16.92	
16.04/16.92	SCC Processor:
16.04/16.92	-> Pairs:
16.04/16.92	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.92	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.92	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.92	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.92	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.92	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.92	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.92	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.92	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.92	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.92	-> Rules:
16.04/16.92	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.92	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.92	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Strongly Connected Components:
16.04/16.93	->->Cycle:
16.04/16.93	->->-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.93	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	->->-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	Reduction Pairs Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
16.04/16.93	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	-> Usable rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Interpretation type:
16.04/16.93	 Linear
16.04/16.93	->Coefficients:
16.04/16.93	 Natural Numbers
16.04/16.93	->Dimension:
16.04/16.93	 1
16.04/16.93	->Bound:
16.04/16.93	 2
16.04/16.93	->Interpretation:
16.04/16.93	 
16.04/16.93	[U11](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.93	[U12](X1,X2) = X1 + 2.X2 + 1
16.04/16.93	[U13](X) = X
16.04/16.93	[U21](X1,X2) = X1 + 2.X2 + 1
16.04/16.93	[U22](X) = X
16.04/16.93	[U31](X1,X2) = X2 + 2
16.04/16.93	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.93	[active](X) = X
16.04/16.93	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.93	[isNat](X) = 2.X + 1
16.04/16.93	[isNatKind](X) = 0
16.04/16.93	[mark](X) = X
16.04/16.93	[plus](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.93	[s](X) = X
16.04/16.93	[0] = 2
16.04/16.93	[fSNonEmpty] = 0
16.04/16.93	[tt] = 0
16.04/16.93	[U11#](X1,X2,X3) = 0
16.04/16.93	[U12#](X1,X2) = 0
16.04/16.93	[U13#](X) = 0
16.04/16.93	[U21#](X1,X2) = 0
16.04/16.93	[U22#](X) = 0
16.04/16.93	[U31#](X1,X2) = 0
16.04/16.93	[U41#](X1,X2,X3) = 0
16.04/16.93	[ACTIVE](X) = X + 2
16.04/16.93	[AND](X1,X2) = 0
16.04/16.93	[ISNAT](X) = 0
16.04/16.93	[ISNATKIND](X) = 0
16.04/16.93	[MARK](X) = 2.X + 2
16.04/16.93	[PLUS](X1,X2) = 0
16.04/16.93	[S](X) = 0
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	SCC Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Strongly Connected Components:
16.04/16.93	->->Cycle:
16.04/16.93	->->-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	->->-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	Reduction Pairs Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	-> Usable rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Interpretation type:
16.04/16.93	 Linear
16.04/16.93	->Coefficients:
16.04/16.93	 Natural Numbers
16.04/16.93	->Dimension:
16.04/16.93	 1
16.04/16.93	->Bound:
16.04/16.93	 2
16.04/16.93	->Interpretation:
16.04/16.93	 
16.04/16.93	[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.04/16.93	[U12](X1,X2) = X1 + 2.X2 + 1
16.04/16.93	[U13](X) = X
16.04/16.93	[U21](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.93	[U22](X) = X
16.04/16.93	[U31](X1,X2) = 2.X2 + 2
16.04/16.93	[U41](X1,X2,X3) = X2 + 2.X3 + 2
16.04/16.93	[active](X) = X
16.04/16.93	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.93	[isNat](X) = 2.X + 1
16.04/16.93	[isNatKind](X) = 0
16.04/16.93	[mark](X) = X
16.04/16.93	[plus](X1,X2) = 2.X1 + X2 + 1
16.04/16.93	[s](X) = X + 1
16.04/16.93	[0] = 2
16.04/16.93	[fSNonEmpty] = 0
16.04/16.93	[tt] = 0
16.04/16.93	[U11#](X1,X2,X3) = 0
16.04/16.93	[U12#](X1,X2) = 0
16.04/16.93	[U13#](X) = 0
16.04/16.93	[U21#](X1,X2) = 0
16.04/16.93	[U22#](X) = 0
16.04/16.93	[U31#](X1,X2) = 0
16.04/16.93	[U41#](X1,X2,X3) = 0
16.04/16.93	[ACTIVE](X) = X + 2
16.04/16.93	[AND](X1,X2) = 0
16.04/16.93	[ISNAT](X) = 0
16.04/16.93	[ISNATKIND](X) = 0
16.04/16.93	[MARK](X) = 2.X + 2
16.04/16.93	[PLUS](X1,X2) = 0
16.04/16.93	[S](X) = 0
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	SCC Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Strongly Connected Components:
16.04/16.93	->->Cycle:
16.04/16.93	->->-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	->->-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	Reduction Pairs Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U13(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	-> Usable rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Interpretation type:
16.04/16.93	 Linear
16.04/16.93	->Coefficients:
16.04/16.93	 Natural Numbers
16.04/16.93	->Dimension:
16.04/16.93	 1
16.04/16.93	->Bound:
16.04/16.93	 2
16.04/16.93	->Interpretation:
16.04/16.93	 
16.04/16.93	[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
16.04/16.93	[U12](X1,X2) = 2.X2 + 2
16.04/16.93	[U13](X) = X + 1
16.04/16.93	[U21](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.93	[U22](X) = X
16.04/16.93	[U31](X1,X2) = 2.X2 + 2
16.04/16.93	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 1
16.04/16.93	[active](X) = X
16.04/16.93	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.93	[isNat](X) = 2.X + 1
16.04/16.93	[isNatKind](X) = 0
16.04/16.93	[mark](X) = X
16.04/16.93	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.93	[s](X) = X
16.04/16.93	[0] = 2
16.04/16.93	[fSNonEmpty] = 0
16.04/16.93	[tt] = 0
16.04/16.93	[U11#](X1,X2,X3) = 0
16.04/16.93	[U12#](X1,X2) = 0
16.04/16.93	[U13#](X) = 0
16.04/16.93	[U21#](X1,X2) = 0
16.04/16.93	[U22#](X) = 0
16.04/16.93	[U31#](X1,X2) = 0
16.04/16.93	[U41#](X1,X2,X3) = 0
16.04/16.93	[ACTIVE](X) = 2.X + 2
16.04/16.93	[AND](X1,X2) = 0
16.04/16.93	[ISNAT](X) = 0
16.04/16.93	[ISNATKIND](X) = 0
16.04/16.93	[MARK](X) = 2.X + 2
16.04/16.93	[PLUS](X1,X2) = 0
16.04/16.93	[S](X) = 0
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	SCC Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Strongly Connected Components:
16.04/16.93	->->Cycle:
16.04/16.93	->->-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	->->-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	Reduction Pairs Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	-> Usable rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Interpretation type:
16.04/16.93	 Linear
16.04/16.93	->Coefficients:
16.04/16.93	 Natural Numbers
16.04/16.93	->Dimension:
16.04/16.93	 1
16.04/16.93	->Bound:
16.04/16.93	 2
16.04/16.93	->Interpretation:
16.04/16.93	 
16.04/16.93	[U11](X1,X2,X3) = 0
16.04/16.93	[U12](X1,X2) = 2
16.04/16.93	[U13](X) = 0
16.04/16.93	[U21](X1,X2) = 1
16.04/16.93	[U22](X) = 2
16.04/16.93	[U31](X1,X2) = 2
16.04/16.93	[U41](X1,X2,X3) = 2
16.04/16.93	[active](X) = 2
16.04/16.93	[and](X1,X2) = 2
16.04/16.93	[isNat](X) = 0
16.04/16.93	[isNatKind](X) = 2
16.04/16.93	[mark](X) = 2
16.04/16.93	[plus](X1,X2) = 2
16.04/16.93	[s](X) = 0
16.04/16.93	[0] = 0
16.04/16.93	[fSNonEmpty] = 0
16.04/16.93	[tt] = 1
16.04/16.93	[U11#](X1,X2,X3) = 0
16.04/16.93	[U12#](X1,X2) = 0
16.04/16.93	[U13#](X) = 0
16.04/16.93	[U21#](X1,X2) = 0
16.04/16.93	[U22#](X) = 0
16.04/16.93	[U31#](X1,X2) = 0
16.04/16.93	[U41#](X1,X2,X3) = 0
16.04/16.93	[ACTIVE](X) = X
16.04/16.93	[AND](X1,X2) = 0
16.04/16.93	[ISNAT](X) = 0
16.04/16.93	[ISNATKIND](X) = 0
16.04/16.93	[MARK](X) = 2
16.04/16.93	[PLUS](X1,X2) = 0
16.04/16.93	[S](X) = 0
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	SCC Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Strongly Connected Components:
16.04/16.93	->->Cycle:
16.04/16.93	->->-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	->->-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	Reduction Pairs Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	-> Usable rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Interpretation type:
16.04/16.93	 Linear
16.04/16.93	->Coefficients:
16.04/16.93	 Natural Numbers
16.04/16.93	->Dimension:
16.04/16.93	 1
16.04/16.93	->Bound:
16.04/16.93	 2
16.04/16.93	->Interpretation:
16.04/16.93	 
16.04/16.93	[U11](X1,X2,X3) = 2.X1 + 2
16.04/16.93	[U12](X1,X2) = 2
16.04/16.93	[U13](X) = 2
16.04/16.93	[U21](X1,X2) = X1 + 2
16.04/16.93	[U22](X) = X
16.04/16.93	[U31](X1,X2) = 2.X2 + 2
16.04/16.93	[U41](X1,X2,X3) = 2.X2 + 2.X3
16.04/16.93	[active](X) = X
16.04/16.93	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.93	[isNat](X) = 2
16.04/16.93	[isNatKind](X) = 0
16.04/16.93	[mark](X) = X
16.04/16.93	[plus](X1,X2) = 2.X1 + 2.X2
16.04/16.93	[s](X) = X
16.04/16.93	[0] = 2
16.04/16.93	[fSNonEmpty] = 0
16.04/16.93	[tt] = 0
16.04/16.93	[U11#](X1,X2,X3) = 0
16.04/16.93	[U12#](X1,X2) = 0
16.04/16.93	[U13#](X) = 0
16.04/16.93	[U21#](X1,X2) = 0
16.04/16.93	[U22#](X) = 0
16.04/16.93	[U31#](X1,X2) = 0
16.04/16.93	[U41#](X1,X2,X3) = 0
16.04/16.93	[ACTIVE](X) = 2.X + 2
16.04/16.93	[AND](X1,X2) = 0
16.04/16.93	[ISNAT](X) = 0
16.04/16.93	[ISNATKIND](X) = 0
16.04/16.93	[MARK](X) = 2.X + 2
16.04/16.93	[PLUS](X1,X2) = 0
16.04/16.93	[S](X) = 0
16.04/16.93	
16.04/16.93	Problem 1.13: 
16.04/16.93	
16.04/16.93	SCC Processor:
16.04/16.93	-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.93	 active(isNatKind(0)) -> mark(tt)
16.04/16.93	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.93	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.93	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.93	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.93	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.93	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.93	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.93	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.93	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.93	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.93	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.93	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.93	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.93	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.93	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.93	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.93	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.93	 mark(0) -> active(0)
16.04/16.93	 mark(tt) -> active(tt)
16.04/16.93	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.93	 s(active(X:S)) -> s(X:S)
16.04/16.93	 s(mark(X:S)) -> s(X:S)
16.04/16.93	->Strongly Connected Components:
16.04/16.93	->->Cycle:
16.04/16.93	->->-> Pairs:
16.04/16.93	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.93	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.93	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.93	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.93	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.93	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.93	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.93	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.93	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.93	->->-> Rules:
16.04/16.93	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.93	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.93	 U13(active(X:S)) -> U13(X:S)
16.04/16.93	 U13(mark(X:S)) -> U13(X:S)
16.04/16.93	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.93	 U22(active(X:S)) -> U22(X:S)
16.04/16.93	 U22(mark(X:S)) -> U22(X:S)
16.04/16.93	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.93	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.93	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.93	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.93	 active(U13(tt)) -> mark(tt)
16.04/16.93	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.93	 active(U22(tt)) -> mark(tt)
16.04/16.93	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.93	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.93	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.93	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.93	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.93	 active(isNat(0)) -> mark(tt)
16.04/16.93	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	
16.04/16.94	Problem 1.13: 
16.04/16.94	
16.04/16.94	Reduction Pairs Processor:
16.04/16.94	-> Pairs:
16.04/16.94	 ACTIVE(and(tt,X:S)) -> MARK(X:S)
16.04/16.94	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	-> Usable rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	->Interpretation type:
16.04/16.94	 Linear
16.04/16.94	->Coefficients:
16.04/16.94	 Natural Numbers
16.04/16.94	->Dimension:
16.04/16.94	 1
16.04/16.94	->Bound:
16.04/16.94	 2
16.04/16.94	->Interpretation:
16.04/16.94	 
16.04/16.94	[U11](X1,X2,X3) = X1 + 2.X3 + 2
16.04/16.94	[U12](X1,X2) = 2.X2 + 2
16.04/16.94	[U13](X) = 2
16.04/16.94	[U21](X1,X2) = 2.X2 + 2
16.04/16.94	[U22](X) = X
16.04/16.94	[U31](X1,X2) = 2.X2 + 2
16.04/16.94	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.94	[active](X) = X
16.04/16.94	[and](X1,X2) = 2.X1 + X2 + 2
16.04/16.94	[isNat](X) = 2.X + 2
16.04/16.94	[isNatKind](X) = 2.X
16.04/16.94	[mark](X) = X
16.04/16.94	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.94	[s](X) = X + 1
16.04/16.94	[0] = 2
16.04/16.94	[fSNonEmpty] = 0
16.04/16.94	[tt] = 2
16.04/16.94	[U11#](X1,X2,X3) = 0
16.04/16.94	[U12#](X1,X2) = 0
16.04/16.94	[U13#](X) = 0
16.04/16.94	[U21#](X1,X2) = 0
16.04/16.94	[U22#](X) = 0
16.04/16.94	[U31#](X1,X2) = 0
16.04/16.94	[U41#](X1,X2,X3) = 0
16.04/16.94	[ACTIVE](X) = X + 2
16.04/16.94	[AND](X1,X2) = 0
16.04/16.94	[ISNAT](X) = 0
16.04/16.94	[ISNATKIND](X) = 0
16.04/16.94	[MARK](X) = X + 2
16.04/16.94	[PLUS](X1,X2) = 0
16.04/16.94	[S](X) = 0
16.04/16.94	
16.04/16.94	Problem 1.13: 
16.04/16.94	
16.04/16.94	SCC Processor:
16.04/16.94	-> Pairs:
16.04/16.94	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	->Strongly Connected Components:
16.04/16.94	->->Cycle:
16.04/16.94	->->-> Pairs:
16.04/16.94	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	->->-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	
16.04/16.94	Problem 1.13: 
16.04/16.94	
16.04/16.94	Reduction Pairs Processor:
16.04/16.94	-> Pairs:
16.04/16.94	 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	-> Usable rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	->Interpretation type:
16.04/16.94	 Linear
16.04/16.94	->Coefficients:
16.04/16.94	 Natural Numbers
16.04/16.94	->Dimension:
16.04/16.94	 1
16.04/16.94	->Bound:
16.04/16.94	 2
16.04/16.94	->Interpretation:
16.04/16.94	 
16.04/16.94	[U11](X1,X2,X3) = X2 + 2.X3 + 1
16.04/16.94	[U12](X1,X2) = X1 + 2.X2
16.04/16.94	[U13](X) = 2.X + 2
16.04/16.94	[U21](X1,X2) = X2
16.04/16.94	[U22](X) = X
16.04/16.94	[U31](X1,X2) = X2 + 2
16.04/16.94	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.94	[active](X) = X
16.04/16.94	[and](X1,X2) = 2.X1 + 2.X2
16.04/16.94	[isNat](X) = X
16.04/16.94	[isNatKind](X) = X
16.04/16.94	[mark](X) = X
16.04/16.94	[plus](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.94	[s](X) = X
16.04/16.94	[0] = 2
16.04/16.94	[fSNonEmpty] = 0
16.04/16.94	[tt] = 2
16.04/16.94	[U11#](X1,X2,X3) = 0
16.04/16.94	[U12#](X1,X2) = 0
16.04/16.94	[U13#](X) = 0
16.04/16.94	[U21#](X1,X2) = 0
16.04/16.94	[U22#](X) = 0
16.04/16.94	[U31#](X1,X2) = 0
16.04/16.94	[U41#](X1,X2,X3) = 0
16.04/16.94	[ACTIVE](X) = 2.X + 2
16.04/16.94	[AND](X1,X2) = 0
16.04/16.94	[ISNAT](X) = 0
16.04/16.94	[ISNATKIND](X) = 0
16.04/16.94	[MARK](X) = 2.X + 2
16.04/16.94	[PLUS](X1,X2) = 0
16.04/16.94	[S](X) = 0
16.04/16.94	
16.04/16.94	Problem 1.13: 
16.04/16.94	
16.04/16.94	SCC Processor:
16.04/16.94	-> Pairs:
16.04/16.94	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	->Strongly Connected Components:
16.04/16.94	->->Cycle:
16.04/16.94	->->-> Pairs:
16.04/16.94	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	->->-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	
16.04/16.94	Problem 1.13: 
16.04/16.94	
16.04/16.94	Reduction Pairs Processor:
16.04/16.94	-> Pairs:
16.04/16.94	 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	-> Usable rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	->Interpretation type:
16.04/16.94	 Linear
16.04/16.94	->Coefficients:
16.04/16.94	 Natural Numbers
16.04/16.94	->Dimension:
16.04/16.94	 1
16.04/16.94	->Bound:
16.04/16.94	 2
16.04/16.94	->Interpretation:
16.04/16.94	 
16.04/16.94	[U11](X1,X2,X3) = 2.X2 + X3 + 2
16.04/16.94	[U12](X1,X2) = 2.X1 + X2
16.04/16.94	[U13](X) = 2
16.04/16.94	[U21](X1,X2) = X2 + 2
16.04/16.94	[U22](X) = X + 1
16.04/16.94	[U31](X1,X2) = 2.X2
16.04/16.94	[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
16.04/16.94	[active](X) = X
16.04/16.94	[and](X1,X2) = 2.X1 + 2.X2 + 2
16.04/16.94	[isNat](X) = X + 1
16.04/16.94	[isNatKind](X) = 2.X
16.04/16.94	[mark](X) = X
16.04/16.94	[plus](X1,X2) = 2.X1 + 2.X2 + 1
16.04/16.94	[s](X) = X + 1
16.04/16.94	[0] = 1
16.04/16.94	[fSNonEmpty] = 0
16.04/16.94	[tt] = 2
16.04/16.94	[U11#](X1,X2,X3) = 0
16.04/16.94	[U12#](X1,X2) = 0
16.04/16.94	[U13#](X) = 0
16.04/16.94	[U21#](X1,X2) = 0
16.04/16.94	[U22#](X) = 0
16.04/16.94	[U31#](X1,X2) = 0
16.04/16.94	[U41#](X1,X2,X3) = 0
16.04/16.94	[ACTIVE](X) = 2.X + 2
16.04/16.94	[AND](X1,X2) = 0
16.04/16.94	[ISNAT](X) = 0
16.04/16.94	[ISNATKIND](X) = 0
16.04/16.94	[MARK](X) = 2.X + 2
16.04/16.94	[PLUS](X1,X2) = 0
16.04/16.94	[S](X) = 0
16.04/16.94	
16.04/16.94	Problem 1.13: 
16.04/16.94	
16.04/16.94	SCC Processor:
16.04/16.94	-> Pairs:
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	->Strongly Connected Components:
16.04/16.94	->->Cycle:
16.04/16.94	->->-> Pairs:
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	->->-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	
16.04/16.94	Problem 1.13: 
16.04/16.94	
16.04/16.94	Subterm Processor:
16.04/16.94	-> Pairs:
16.04/16.94	 MARK(U22(X:S)) -> MARK(X:S)
16.04/16.94	 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
16.04/16.94	 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
16.04/16.94	 MARK(s(X:S)) -> MARK(X:S)
16.04/16.94	-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	->Projection:
16.04/16.94	 pi(MARK) = 1
16.04/16.94	
16.04/16.94	Problem 1.13: 
16.04/16.94	
16.04/16.94	SCC Processor:
16.04/16.94	-> Pairs:
16.04/16.94	 Empty
16.04/16.94	-> Rules:
16.04/16.94	 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
16.04/16.94	 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
16.04/16.94	 U13(active(X:S)) -> U13(X:S)
16.04/16.94	 U13(mark(X:S)) -> U13(X:S)
16.04/16.94	 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
16.04/16.94	 U22(active(X:S)) -> U22(X:S)
16.04/16.94	 U22(mark(X:S)) -> U22(X:S)
16.04/16.94	 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
16.04/16.94	 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
16.04/16.94	 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
16.04/16.94	 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
16.04/16.94	 active(U13(tt)) -> mark(tt)
16.04/16.94	 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
16.04/16.94	 active(U22(tt)) -> mark(tt)
16.04/16.94	 active(U31(tt,N:S)) -> mark(N:S)
16.04/16.94	 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
16.04/16.94	 active(and(tt,X:S)) -> mark(X:S)
16.04/16.94	 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
16.04/16.94	 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
16.04/16.94	 active(isNat(0)) -> mark(tt)
16.04/16.94	 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
16.04/16.94	 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
16.04/16.94	 active(isNatKind(0)) -> mark(tt)
16.04/16.94	 active(plus(N:S,s(M:S))) -> mark(U41(and(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S))),M:S,N:S))
16.04/16.94	 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
16.04/16.94	 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
16.04/16.94	 isNat(active(X:S)) -> isNat(X:S)
16.04/16.94	 isNat(mark(X:S)) -> isNat(X:S)
16.04/16.94	 isNatKind(active(X:S)) -> isNatKind(X:S)
16.04/16.94	 isNatKind(mark(X:S)) -> isNatKind(X:S)
16.04/16.94	 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
16.04/16.94	 mark(U13(X:S)) -> active(U13(mark(X:S)))
16.04/16.94	 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
16.04/16.94	 mark(U22(X:S)) -> active(U22(mark(X:S)))
16.04/16.94	 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
16.04/16.94	 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
16.04/16.94	 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
16.04/16.94	 mark(isNat(X:S)) -> active(isNat(X:S))
16.04/16.94	 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
16.04/16.94	 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
16.04/16.94	 mark(s(X:S)) -> active(s(mark(X:S)))
16.04/16.94	 mark(0) -> active(0)
16.04/16.94	 mark(tt) -> active(tt)
16.04/16.94	 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
16.04/16.94	 s(active(X:S)) -> s(X:S)
16.04/16.94	 s(mark(X:S)) -> s(X:S)
16.04/16.94	->Strongly Connected Components:
16.04/16.94	 There is no strongly connected component
16.04/16.94	
16.04/16.94	The problem is finite.
16.04/16.94	EOF