YES

Problem 1: 

(VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S X3:S)
(RULES
U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
U13(active(X:S)) -> U13(X:S)
U13(mark(X:S)) -> U13(X:S)
U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
U22(active(X:S)) -> U22(X:S)
U22(mark(X:S)) -> U22(X:S)
U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
active(U13(tt)) -> mark(tt)
active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
active(U22(tt)) -> mark(tt)
active(U31(tt,N:S)) -> mark(N:S)
active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
active(and(tt,X:S)) -> mark(X:S)
active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
active(isNat(0)) -> mark(tt)
active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
active(isNatKind(0)) -> mark(tt)
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))
active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
and(active(X1:S),X2:S) -> and(X1:S,X2:S)
and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
isNat(active(X:S)) -> isNat(X:S)
isNat(mark(X:S)) -> isNat(X:S)
isNatKind(active(X:S)) -> isNatKind(X:S)
isNatKind(mark(X:S)) -> isNatKind(X:S)
mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
mark(U13(X:S)) -> active(U13(mark(X:S)))
mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
mark(U22(X:S)) -> active(U22(mark(X:S)))
mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
mark(isNat(X:S)) -> active(isNat(X:S))
mark(isNatKind(X:S)) -> active(isNatKind(X:S))
mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
mark(s(X:S)) -> active(s(mark(X:S)))
mark(0) -> active(0)
mark(tt) -> active(tt)
plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
s(active(X:S)) -> s(X:S)
s(mark(X:S)) -> s(X:S)
) 
(STRATEGY INNERMOST)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U12#(active(X1:S),X2:S) -> U12#(X1:S,X2:S)
 U12#(mark(X1:S),X2:S) -> U12#(X1:S,X2:S)
 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
 U13#(active(X:S)) -> U13#(X:S)
 U13#(mark(X:S)) -> U13#(X:S)
 U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S)
 U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S)
 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
 U22#(active(X:S)) -> U22#(X:S)
 U22#(mark(X:S)) -> U22#(X:S)
 U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S)
 U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S)
 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
 U41#(active(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(mark(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 ACTIVE(U11(tt,V1:S,V2:S)) -> U12#(isNat(V1:S),V2:S)
 ACTIVE(U11(tt,V1:S,V2:S)) -> ISNAT(V1:S)
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> U13#(isNat(V2:S))
 ACTIVE(U12(tt,V2:S)) -> ISNAT(V2:S)
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U13(tt)) -> MARK(tt)
 ACTIVE(U21(tt,V1:S)) -> U22#(isNat(V1:S))
 ACTIVE(U21(tt,V1:S)) -> ISNAT(V1:S)
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(U22(tt)) -> MARK(tt)
 ACTIVE(U31(tt,N:S)) -> MARK(N:S)
 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(U41(tt,M:S,N:S)) -> PLUS(N:S,M:S)
 ACTIVE(U41(tt,M:S,N:S)) -> S(plus(N:S,M:S))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> U11#(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> AND(isNatKind(V1:S),isNatKind(V2:S))
 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V1:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V2:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> U21#(isNatKind(V1:S),V1:S)
 ACTIVE(isNat(s(V1:S))) -> ISNATKIND(V1:S)
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNat(0)) -> MARK(tt)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> AND(isNatKind(V1:S),isNatKind(V2:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V1:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V2:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> ISNATKIND(V1:S)
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 ACTIVE(isNatKind(0)) -> MARK(tt)
 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)
 ACTIVE(plus(N:S,s(M:S))) -> AND(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S)))
 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(M:S),isNatKind(M:S))
 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(N:S),isNatKind(N:S))
 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S)
 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(N:S)
 ACTIVE(plus(N:S,s(M:S))) -> ISNATKIND(M:S)
 ACTIVE(plus(N:S,s(M:S))) -> ISNATKIND(N:S)
 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))
 ACTIVE(plus(N:S,0)) -> U31#(and(isNat(N:S),isNatKind(N:S)),N:S)
 ACTIVE(plus(N:S,0)) -> AND(isNat(N:S),isNatKind(N:S))
 ACTIVE(plus(N:S,0)) -> ISNAT(N:S)
 ACTIVE(plus(N:S,0)) -> ISNATKIND(N:S)
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
 ISNAT(active(X:S)) -> ISNAT(X:S)
 ISNAT(mark(X:S)) -> ISNAT(X:S)
 ISNATKIND(active(X:S)) -> ISNATKIND(X:S)
 ISNATKIND(mark(X:S)) -> ISNATKIND(X:S)
 MARK(U11(X1:S,X2:S,X3:S)) -> U11#(mark(X1:S),X2:S,X3:S)
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> U12#(mark(X1:S),X2:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> U13#(mark(X:S))
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> U21#(mark(X1:S),X2:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> U22#(mark(X:S))
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> U31#(mark(X1:S),X2:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> U41#(mark(X1:S),X2:S,X3:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S)
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S))
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> S(mark(X:S))
 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U12#(active(X1:S),X2:S) -> U12#(X1:S,X2:S)
 U12#(mark(X1:S),X2:S) -> U12#(X1:S,X2:S)
 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
 U13#(active(X:S)) -> U13#(X:S)
 U13#(mark(X:S)) -> U13#(X:S)
 U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S)
 U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S)
 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
 U22#(active(X:S)) -> U22#(X:S)
 U22#(mark(X:S)) -> U22#(X:S)
 U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S)
 U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S)
 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
 U41#(active(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(mark(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 ACTIVE(U11(tt,V1:S,V2:S)) -> U12#(isNat(V1:S),V2:S)
 ACTIVE(U11(tt,V1:S,V2:S)) -> ISNAT(V1:S)
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> U13#(isNat(V2:S))
 ACTIVE(U12(tt,V2:S)) -> ISNAT(V2:S)
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U13(tt)) -> MARK(tt)
 ACTIVE(U21(tt,V1:S)) -> U22#(isNat(V1:S))
 ACTIVE(U21(tt,V1:S)) -> ISNAT(V1:S)
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(U22(tt)) -> MARK(tt)
 ACTIVE(U31(tt,N:S)) -> MARK(N:S)
 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(U41(tt,M:S,N:S)) -> PLUS(N:S,M:S)
 ACTIVE(U41(tt,M:S,N:S)) -> S(plus(N:S,M:S))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> U11#(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> AND(isNatKind(V1:S),isNatKind(V2:S))
 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V1:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> ISNATKIND(V2:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> U21#(isNatKind(V1:S),V1:S)
 ACTIVE(isNat(s(V1:S))) -> ISNATKIND(V1:S)
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNat(0)) -> MARK(tt)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> AND(isNatKind(V1:S),isNatKind(V2:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V1:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> ISNATKIND(V2:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> ISNATKIND(V1:S)
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 ACTIVE(isNatKind(0)) -> MARK(tt)
 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)
 ACTIVE(plus(N:S,s(M:S))) -> AND(and(isNat(M:S),isNatKind(M:S)),and(isNat(N:S),isNatKind(N:S)))
 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(M:S),isNatKind(M:S))
 ACTIVE(plus(N:S,s(M:S))) -> AND(isNat(N:S),isNatKind(N:S))
 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(M:S)
 ACTIVE(plus(N:S,s(M:S))) -> ISNAT(N:S)
 ACTIVE(plus(N:S,s(M:S))) -> ISNATKIND(M:S)
 ACTIVE(plus(N:S,s(M:S))) -> ISNATKIND(N:S)
 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))
 ACTIVE(plus(N:S,0)) -> U31#(and(isNat(N:S),isNatKind(N:S)),N:S)
 ACTIVE(plus(N:S,0)) -> AND(isNat(N:S),isNatKind(N:S))
 ACTIVE(plus(N:S,0)) -> ISNAT(N:S)
 ACTIVE(plus(N:S,0)) -> ISNATKIND(N:S)
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
 ISNAT(active(X:S)) -> ISNAT(X:S)
 ISNAT(mark(X:S)) -> ISNAT(X:S)
 ISNATKIND(active(X:S)) -> ISNATKIND(X:S)
 ISNATKIND(mark(X:S)) -> ISNATKIND(X:S)
 MARK(U11(X1:S,X2:S,X3:S)) -> U11#(mark(X1:S),X2:S,X3:S)
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> U12#(mark(X1:S),X2:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> U13#(mark(X:S))
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> U21#(mark(X1:S),X2:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> U22#(mark(X:S))
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> U31#(mark(X1:S),X2:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> U41#(mark(X1:S),X2:S,X3:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> AND(mark(X1:S),X2:S)
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(plus(X1:S,X2:S)) -> PLUS(mark(X1:S),mark(X2:S))
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
 MARK(s(X:S)) -> S(mark(X:S))
 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 ISNATKIND(active(X:S)) -> ISNATKIND(X:S)
 ISNATKIND(mark(X:S)) -> ISNATKIND(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 ISNAT(active(X:S)) -> ISNAT(X:S)
 ISNAT(mark(X:S)) -> ISNAT(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 U41#(active(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(mark(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S)
 U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S)
 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 U22#(active(X:S)) -> U22#(X:S)
 U22#(mark(X:S)) -> U22#(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S)
 U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S)
 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 U13#(active(X:S)) -> U13#(X:S)
 U13#(mark(X:S)) -> U13#(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 U12#(active(X1:S),X2:S) -> U12#(X1:S,X2:S)
 U12#(mark(X1:S),X2:S) -> U12#(X1:S,X2:S)
 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(U31(tt,N:S)) -> MARK(N:S)
 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 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))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)


The problem is decomposed in 13 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 S(active(X:S)) -> S(X:S)
 S(mark(X:S)) -> S(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(S) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 PLUS(active(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(mark(X1:S),X2:S) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(PLUS) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.2: 

Subterm Processor:
-> Pairs:
 PLUS(X1:S,active(X2:S)) -> PLUS(X1:S,X2:S)
 PLUS(X1:S,mark(X2:S)) -> PLUS(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(PLUS) = 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> Pairs:
 ISNATKIND(active(X:S)) -> ISNATKIND(X:S)
 ISNATKIND(mark(X:S)) -> ISNATKIND(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(ISNATKIND) = 1

Problem 1.3: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Subterm Processor:
-> Pairs:
 ISNAT(active(X:S)) -> ISNAT(X:S)
 ISNAT(mark(X:S)) -> ISNAT(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(ISNAT) = 1

Problem 1.4: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.5: 

Subterm Processor:
-> Pairs:
 AND(active(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(mark(X1:S),X2:S) -> AND(X1:S,X2:S)
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(AND) = 1

Problem 1.5: 

SCC Processor:
-> Pairs:
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.5: 

Subterm Processor:
-> Pairs:
 AND(X1:S,active(X2:S)) -> AND(X1:S,X2:S)
 AND(X1:S,mark(X2:S)) -> AND(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(AND) = 2

Problem 1.5: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.6: 

Subterm Processor:
-> Pairs:
 U41#(active(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(mark(X1:S),X2:S,X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U41#) = 1

Problem 1.6: 

SCC Processor:
-> Pairs:
 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.6: 

Subterm Processor:
-> Pairs:
 U41#(X1:S,active(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,mark(X2:S),X3:S) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U41#) = 2

Problem 1.6: 

SCC Processor:
-> Pairs:
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.6: 

Subterm Processor:
-> Pairs:
 U41#(X1:S,X2:S,active(X3:S)) -> U41#(X1:S,X2:S,X3:S)
 U41#(X1:S,X2:S,mark(X3:S)) -> U41#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U41#) = 3

Problem 1.6: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.7: 

Subterm Processor:
-> Pairs:
 U31#(active(X1:S),X2:S) -> U31#(X1:S,X2:S)
 U31#(mark(X1:S),X2:S) -> U31#(X1:S,X2:S)
 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U31#) = 1

Problem 1.7: 

SCC Processor:
-> Pairs:
 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.7: 

Subterm Processor:
-> Pairs:
 U31#(X1:S,active(X2:S)) -> U31#(X1:S,X2:S)
 U31#(X1:S,mark(X2:S)) -> U31#(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U31#) = 2

Problem 1.7: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.8: 

Subterm Processor:
-> Pairs:
 U22#(active(X:S)) -> U22#(X:S)
 U22#(mark(X:S)) -> U22#(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U22#) = 1

Problem 1.8: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.9: 

Subterm Processor:
-> Pairs:
 U21#(active(X1:S),X2:S) -> U21#(X1:S,X2:S)
 U21#(mark(X1:S),X2:S) -> U21#(X1:S,X2:S)
 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U21#) = 1

Problem 1.9: 

SCC Processor:
-> Pairs:
 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.9: 

Subterm Processor:
-> Pairs:
 U21#(X1:S,active(X2:S)) -> U21#(X1:S,X2:S)
 U21#(X1:S,mark(X2:S)) -> U21#(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U21#) = 2

Problem 1.9: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.10: 

Subterm Processor:
-> Pairs:
 U13#(active(X:S)) -> U13#(X:S)
 U13#(mark(X:S)) -> U13#(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U13#) = 1

Problem 1.10: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.11: 

Subterm Processor:
-> Pairs:
 U12#(active(X1:S),X2:S) -> U12#(X1:S,X2:S)
 U12#(mark(X1:S),X2:S) -> U12#(X1:S,X2:S)
 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U12#) = 1

Problem 1.11: 

SCC Processor:
-> Pairs:
 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.11: 

Subterm Processor:
-> Pairs:
 U12#(X1:S,active(X2:S)) -> U12#(X1:S,X2:S)
 U12#(X1:S,mark(X2:S)) -> U12#(X1:S,X2:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U12#) = 2

Problem 1.11: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.12: 

Subterm Processor:
-> Pairs:
 U11#(active(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(mark(X1:S),X2:S,X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U11#) = 1

Problem 1.12: 

SCC Processor:
-> Pairs:
 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.12: 

Subterm Processor:
-> Pairs:
 U11#(X1:S,active(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,mark(X2:S),X3:S) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U11#) = 2

Problem 1.12: 

SCC Processor:
-> Pairs:
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.12: 

Subterm Processor:
-> Pairs:
 U11#(X1:S,X2:S,active(X3:S)) -> U11#(X1:S,X2:S,X3:S)
 U11#(X1:S,X2:S,mark(X3:S)) -> U11#(X1:S,X2:S,X3:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(U11#) = 3

Problem 1.12: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(U31(tt,N:S)) -> MARK(N:S)
 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 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))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1
[U12](X1,X2) = 2.X1
[U13](X) = X
[U21](X1,X2) = X1
[U22](X) = 2.X
[U31](X1,X2) = 2.X1 + 2.X2 + 2
[U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 0
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2
[s](X) = X
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 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))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 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))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(U41(tt,M:S,N:S)) -> MARK(s(plus(N:S,M:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 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))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1
[U12](X1,X2) = 2.X1
[U13](X) = 2.X
[U21](X1,X2) = 2.X1
[U22](X) = X
[U31](X1,X2) = 2.X1 + 2.X2
[U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 0
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2
[s](X) = X + 1
[0] = 0
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 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))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 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))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 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))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1
[U12](X1,X2) = 2.X1
[U13](X) = 2.X
[U21](X1,X2) = X1
[U22](X) = X
[U31](X1,X2) = 2.X1 + 2.X2 + 1
[U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 0
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[0] = 0
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 ACTIVE(plus(N:S,0)) -> MARK(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1
[U12](X1,X2) = 2.X1
[U13](X) = 2.X
[U21](X1,X2) = 2.X1
[U22](X) = X
[U31](X1,X2) = 2.X1 + 2.X2 + 1
[U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + X2
[isNat](X) = 0
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[0] = 1
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> ACTIVE(U13(mark(X:S)))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 1
[U12](X1,X2) = 1
[U13](X) = 0
[U21](X1,X2) = 1
[U22](X) = 1
[U31](X1,X2) = 1
[U41](X1,X2,X3) = 1
[active](X) = 2
[and](X1,X2) = 1
[isNat](X) = 1
[isNatKind](X) = 1
[mark](X) = 2
[plus](X1,X2) = 0
[s](X) = 1
[0] = 1
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> ACTIVE(U22(mark(X:S)))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 1
[U12](X1,X2) = 1
[U13](X) = 2
[U21](X1,X2) = 1
[U22](X) = 0
[U31](X1,X2) = 1
[U41](X1,X2,X3) = 1
[active](X) = 2
[and](X1,X2) = 1
[isNat](X) = 1
[isNatKind](X) = 1
[mark](X) = 2
[plus](X1,X2) = 0
[s](X) = 1
[0] = 2
[fSNonEmpty] = 0
[tt] = 2
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> ACTIVE(U31(mark(X1:S),X2:S))
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 1
[U12](X1,X2) = 1
[U13](X) = 2
[U21](X1,X2) = 1
[U22](X) = 2
[U31](X1,X2) = 0
[U41](X1,X2,X3) = 0
[active](X) = 2
[and](X1,X2) = 1
[isNat](X) = 1
[isNatKind](X) = 1
[mark](X) = 2
[plus](X1,X2) = 1
[s](X) = 1
[0] = 0
[fSNonEmpty] = 0
[tt] = 2
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U31(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1
[U12](X1,X2) = 2.X1
[U13](X) = 2.X
[U21](X1,X2) = 2.X1
[U22](X) = 2.X
[U31](X1,X2) = X1 + X2 + 1
[U41](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 0
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 1
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 1
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> ACTIVE(U41(mark(X1:S),X2:S,X3:S))
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1
[U12](X1,X2) = 2.X1
[U13](X) = 2.X
[U21](X1,X2) = X1
[U22](X) = 2.X
[U31](X1,X2) = X1 + 2.X2 + 2
[U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 0
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(U41(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1
[U12](X1,X2) = X1
[U13](X) = X
[U21](X1,X2) = X1
[U22](X) = X
[U31](X1,X2) = 2.X1 + 2.X2 + 2
[U41](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 0
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U11(tt,V1:S,V2:S)) -> MARK(U12(isNat(V1:S),V2:S))
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2) = X1 + 2.X2 + 1
[U13](X) = X + 1
[U21](X1,X2) = 2.X1 + 2.X2 + 2
[U22](X) = X + 2
[U31](X1,X2) = 2.X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U12(tt,V2:S)) -> MARK(U13(isNat(V2:S)))
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2) = X1 + 2.X2 + 2
[U13](X) = X + 1
[U21](X1,X2) = X1 + 2.X2 + 2
[U22](X) = X + 2
[U31](X1,X2) = 2.X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[0] = 1
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(U21(tt,V1:S)) -> MARK(U22(isNat(V1:S)))
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2) = X1 + 2.X2 + 2
[U13](X) = X + 2
[U21](X1,X2) = 2.X1 + 2.X2 + 2
[U22](X) = X + 1
[U31](X1,X2) = X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(plus(V1:S,V2:S))) -> MARK(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2) = X1 + 2.X2 + 1
[U13](X) = X
[U21](X1,X2) = X1 + 2.X2 + 2
[U22](X) = X + 1
[U31](X1,X2) = 2.X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X + 1
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[0] = 1
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNat(s(V1:S))) -> MARK(U21(isNatKind(V1:S),V1:S))
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U12](X1,X2) = X1 + 2.X2
[U13](X) = X
[U21](X1,X2) = 2.X1 + 2.X2 + 2
[U22](X) = X + 1
[U31](X1,X2) = 2.X2 + 1
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = X1 + 2.X2
[isNat](X) = 2.X
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2
[s](X) = X + 2
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNat(X:S)) -> ACTIVE(isNat(X:S))
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> ACTIVE(U11(mark(X1:S),X2:S,X3:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2) = X1 + 2.X2 + 1
[U13](X) = X
[U21](X1,X2) = 2.X1 + 2.X2 + 1
[U22](X) = X
[U31](X1,X2) = X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X + 1
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U11(X1:S,X2:S,X3:S)) -> MARK(X1:S)
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2) = X1 + 2.X2 + 1
[U13](X) = X
[U21](X1,X2) = 2.X1 + 2.X2 + 1
[U22](X) = X
[U31](X1,X2) = 2.X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X + 1
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U12(X1:S,X2:S)) -> ACTIVE(U12(mark(X1:S),X2:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U12](X1,X2) = X1 + 2.X2 + 1
[U13](X) = X
[U21](X1,X2) = X1 + 2.X2 + 1
[U22](X) = X
[U31](X1,X2) = X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X + 1
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U12(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2) = X1 + 2.X2 + 1
[U13](X) = X
[U21](X1,X2) = 2.X1 + 2.X2 + 2
[U22](X) = X
[U31](X1,X2) = 2.X2 + 2
[U41](X1,X2,X3) = X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X + 1
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + X2 + 1
[s](X) = X + 1
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U13(X:S)) -> MARK(X:S)
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2) = 2.X2 + 2
[U13](X) = X + 1
[U21](X1,X2) = 2.X1 + 2.X2 + 1
[U22](X) = X
[U31](X1,X2) = 2.X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 1
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2.X + 1
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U21(X1:S,X2:S)) -> ACTIVE(U21(mark(X1:S),X2:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 0
[U12](X1,X2) = 2
[U13](X) = 0
[U21](X1,X2) = 1
[U22](X) = 2
[U31](X1,X2) = 2
[U41](X1,X2,X3) = 2
[active](X) = 2
[and](X1,X2) = 2
[isNat](X) = 0
[isNatKind](X) = 2
[mark](X) = 2
[plus](X1,X2) = 2
[s](X) = 0
[0] = 0
[fSNonEmpty] = 0
[tt] = 1
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = X
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U21(X1:S,X2:S)) -> MARK(X1:S)
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2
[U12](X1,X2) = 2
[U13](X) = 2
[U21](X1,X2) = X1 + 2
[U22](X) = X
[U31](X1,X2) = 2.X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = 2
[isNatKind](X) = 0
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2
[s](X) = X
[0] = 2
[fSNonEmpty] = 0
[tt] = 0
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(and(tt,X:S)) -> MARK(X:S)
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1 + 2.X3 + 2
[U12](X1,X2) = 2.X2 + 2
[U13](X) = 2
[U21](X1,X2) = 2.X2 + 2
[U22](X) = X
[U31](X1,X2) = 2.X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + X2 + 2
[isNat](X) = 2.X + 2
[isNatKind](X) = 2.X
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[0] = 2
[fSNonEmpty] = 0
[tt] = 2
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(isNatKind(plus(V1:S,V2:S))) -> MARK(and(isNatKind(V1:S),isNatKind(V2:S)))
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X2 + 2.X3 + 1
[U12](X1,X2) = X1 + 2.X2
[U13](X) = 2.X + 2
[U21](X1,X2) = X2
[U22](X) = X
[U31](X1,X2) = X2 + 2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2
[isNat](X) = X
[isNatKind](X) = X
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[0] = 2
[fSNonEmpty] = 0
[tt] = 2
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Reduction Pairs Processor:
-> Pairs:
 ACTIVE(isNatKind(s(V1:S))) -> MARK(isNatKind(V1:S))
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
-> Usable rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X2 + X3 + 2
[U12](X1,X2) = 2.X1 + X2
[U13](X) = 2
[U21](X1,X2) = X2 + 2
[U22](X) = X + 1
[U31](X1,X2) = 2.X2
[U41](X1,X2,X3) = 2.X2 + 2.X3 + 2
[active](X) = X
[and](X1,X2) = 2.X1 + 2.X2 + 2
[isNat](X) = X + 1
[isNatKind](X) = 2.X
[mark](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[0] = 1
[fSNonEmpty] = 0
[tt] = 2
[U11#](X1,X2,X3) = 0
[U12#](X1,X2) = 0
[U13#](X) = 0
[U21#](X1,X2) = 0
[U22#](X) = 0
[U31#](X1,X2) = 0
[U41#](X1,X2,X3) = 0
[ACTIVE](X) = 2.X + 2
[AND](X1,X2) = 0
[ISNAT](X) = 0
[ISNATKIND](X) = 0
[MARK](X) = 2.X + 2
[PLUS](X1,X2) = 0
[S](X) = 0

Problem 1.13: 

SCC Processor:
-> Pairs:
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> ACTIVE(and(mark(X1:S),X2:S))
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(isNatKind(X:S)) -> ACTIVE(isNatKind(X:S))
 MARK(plus(X1:S,X2:S)) -> ACTIVE(plus(mark(X1:S),mark(X2:S)))
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> ACTIVE(s(mark(X:S)))
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> MARK(X:S)
->->-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)

Problem 1.13: 

Subterm Processor:
-> Pairs:
 MARK(U22(X:S)) -> MARK(X:S)
 MARK(and(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X1:S)
 MARK(plus(X1:S,X2:S)) -> MARK(X2:S)
 MARK(s(X:S)) -> MARK(X:S)
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Projection:
 pi(MARK) = 1

Problem 1.13: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 U11(active(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(mark(X1:S),X2:S,X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,active(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,mark(X2:S),X3:S) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,active(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U11(X1:S,X2:S,mark(X3:S)) -> U11(X1:S,X2:S,X3:S)
 U12(active(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(mark(X1:S),X2:S) -> U12(X1:S,X2:S)
 U12(X1:S,active(X2:S)) -> U12(X1:S,X2:S)
 U12(X1:S,mark(X2:S)) -> U12(X1:S,X2:S)
 U13(active(X:S)) -> U13(X:S)
 U13(mark(X:S)) -> U13(X:S)
 U21(active(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(mark(X1:S),X2:S) -> U21(X1:S,X2:S)
 U21(X1:S,active(X2:S)) -> U21(X1:S,X2:S)
 U21(X1:S,mark(X2:S)) -> U21(X1:S,X2:S)
 U22(active(X:S)) -> U22(X:S)
 U22(mark(X:S)) -> U22(X:S)
 U31(active(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(mark(X1:S),X2:S) -> U31(X1:S,X2:S)
 U31(X1:S,active(X2:S)) -> U31(X1:S,X2:S)
 U31(X1:S,mark(X2:S)) -> U31(X1:S,X2:S)
 U41(active(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(mark(X1:S),X2:S,X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,active(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,mark(X2:S),X3:S) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,active(X3:S)) -> U41(X1:S,X2:S,X3:S)
 U41(X1:S,X2:S,mark(X3:S)) -> U41(X1:S,X2:S,X3:S)
 active(U11(tt,V1:S,V2:S)) -> mark(U12(isNat(V1:S),V2:S))
 active(U12(tt,V2:S)) -> mark(U13(isNat(V2:S)))
 active(U13(tt)) -> mark(tt)
 active(U21(tt,V1:S)) -> mark(U22(isNat(V1:S)))
 active(U22(tt)) -> mark(tt)
 active(U31(tt,N:S)) -> mark(N:S)
 active(U41(tt,M:S,N:S)) -> mark(s(plus(N:S,M:S)))
 active(and(tt,X:S)) -> mark(X:S)
 active(isNat(plus(V1:S,V2:S))) -> mark(U11(and(isNatKind(V1:S),isNatKind(V2:S)),V1:S,V2:S))
 active(isNat(s(V1:S))) -> mark(U21(isNatKind(V1:S),V1:S))
 active(isNat(0)) -> mark(tt)
 active(isNatKind(plus(V1:S,V2:S))) -> mark(and(isNatKind(V1:S),isNatKind(V2:S)))
 active(isNatKind(s(V1:S))) -> mark(isNatKind(V1:S))
 active(isNatKind(0)) -> mark(tt)
 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))
 active(plus(N:S,0)) -> mark(U31(and(isNat(N:S),isNatKind(N:S)),N:S))
 and(active(X1:S),X2:S) -> and(X1:S,X2:S)
 and(mark(X1:S),X2:S) -> and(X1:S,X2:S)
 and(X1:S,active(X2:S)) -> and(X1:S,X2:S)
 and(X1:S,mark(X2:S)) -> and(X1:S,X2:S)
 isNat(active(X:S)) -> isNat(X:S)
 isNat(mark(X:S)) -> isNat(X:S)
 isNatKind(active(X:S)) -> isNatKind(X:S)
 isNatKind(mark(X:S)) -> isNatKind(X:S)
 mark(U11(X1:S,X2:S,X3:S)) -> active(U11(mark(X1:S),X2:S,X3:S))
 mark(U12(X1:S,X2:S)) -> active(U12(mark(X1:S),X2:S))
 mark(U13(X:S)) -> active(U13(mark(X:S)))
 mark(U21(X1:S,X2:S)) -> active(U21(mark(X1:S),X2:S))
 mark(U22(X:S)) -> active(U22(mark(X:S)))
 mark(U31(X1:S,X2:S)) -> active(U31(mark(X1:S),X2:S))
 mark(U41(X1:S,X2:S,X3:S)) -> active(U41(mark(X1:S),X2:S,X3:S))
 mark(and(X1:S,X2:S)) -> active(and(mark(X1:S),X2:S))
 mark(isNat(X:S)) -> active(isNat(X:S))
 mark(isNatKind(X:S)) -> active(isNatKind(X:S))
 mark(plus(X1:S,X2:S)) -> active(plus(mark(X1:S),mark(X2:S)))
 mark(s(X:S)) -> active(s(mark(X:S)))
 mark(0) -> active(0)
 mark(tt) -> active(tt)
 plus(active(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(mark(X1:S),X2:S) -> plus(X1:S,X2:S)
 plus(X1:S,active(X2:S)) -> plus(X1:S,X2:S)
 plus(X1:S,mark(X2:S)) -> plus(X1:S,X2:S)
 s(active(X:S)) -> s(X:S)
 s(mark(X:S)) -> s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.