YES

Problem 1: 

(VAR v_NonEmpty:S M:S N:S V1:S V2:S X:S X1:S X2:S)
(RULES
0 -> n__0
U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
U16(tt) -> tt
U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
U23(tt) -> tt
U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
U32(tt) -> tt
U41(tt) -> tt
U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
U52(tt,N:S) -> activate(N:S)
U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
activate(n__0) -> 0
activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
activate(n__s(X:S)) -> s(activate(X:S))
activate(X:S) -> X:S
isNat(n__0) -> tt
isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
isNatKind(n__0) -> tt
isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
plus(N:S,0) -> U51(isNat(N:S),N:S)
plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
s(X:S) -> n__s(X:S)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U11#(tt,V1:S,V2:S) -> U12#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U12#(tt,V1:S,V2:S) -> U13#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U12#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U13#(tt,V1:S,V2:S) -> U14#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U13#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U14#(tt,V1:S,V2:S) -> U15#(isNat(activate(V1:S)),activate(V2:S))
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U14#(tt,V1:S,V2:S) -> ISNAT(activate(V1:S))
 U15#(tt,V2:S) -> U16#(isNat(activate(V2:S)))
 U15#(tt,V2:S) -> ACTIVATE(V2:S)
 U15#(tt,V2:S) -> ISNAT(activate(V2:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> U23#(isNat(activate(V1:S)))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> U32#(isNatKind(activate(V2:S)))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 U64#(tt,M:S,N:S) -> S(plus(activate(N:S),activate(M:S)))
 ACTIVATE(n__0) -> 0#
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> S(activate(X:S))
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> U41#(isNatKind(activate(V1:S)))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(tt,V1:S,V2:S) -> U12#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U12#(tt,V1:S,V2:S) -> U13#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U12#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U13#(tt,V1:S,V2:S) -> U14#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U13#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U14#(tt,V1:S,V2:S) -> U15#(isNat(activate(V1:S)),activate(V2:S))
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U14#(tt,V1:S,V2:S) -> ISNAT(activate(V1:S))
 U15#(tt,V2:S) -> U16#(isNat(activate(V2:S)))
 U15#(tt,V2:S) -> ACTIVATE(V2:S)
 U15#(tt,V2:S) -> ISNAT(activate(V2:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> U23#(isNat(activate(V1:S)))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> U32#(isNatKind(activate(V2:S)))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 U64#(tt,M:S,N:S) -> S(plus(activate(N:S),activate(M:S)))
 ACTIVATE(n__0) -> 0#
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ACTIVATE(n__s(X:S)) -> S(activate(X:S))
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> U41#(isNatKind(activate(V1:S)))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(tt,V1:S,V2:S) -> U12#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U12#(tt,V1:S,V2:S) -> U13#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U12#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U13#(tt,V1:S,V2:S) -> U14#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U13#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U14#(tt,V1:S,V2:S) -> U15#(isNat(activate(V1:S)),activate(V2:S))
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U14#(tt,V1:S,V2:S) -> ISNAT(activate(V1:S))
 U15#(tt,V2:S) -> ACTIVATE(V2:S)
 U15#(tt,V2:S) -> ISNAT(activate(V2:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U11#(tt,V1:S,V2:S) -> U12#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U12#(tt,V1:S,V2:S) -> U13#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U12#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U13#(tt,V1:S,V2:S) -> U14#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U13#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U14#(tt,V1:S,V2:S) -> U15#(isNat(activate(V1:S)),activate(V2:S))
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U14#(tt,V1:S,V2:S) -> ISNAT(activate(V1:S))
 U15#(tt,V2:S) -> ACTIVATE(V2:S)
 U15#(tt,V2:S) -> ISNAT(activate(V2:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2,X3) = X1 + 2.X3 + 2
[U13](X1,X2,X3) = 2.X3 + 2
[U14](X1,X2,X3) = 2.X3 + 2
[U15](X1,X2) = 2.X2 + 2
[U16](X) = X + 2
[U21](X1,X2) = 2.X2
[U22](X1,X2) = 2.X2
[U23](X) = X
[U31](X1,X2) = 2.X1 + 2.X2 + 1
[U32](X) = X + 2
[U41](X) = 2
[U51](X1,X2) = 2.X2 + 2
[U52](X1,X2) = 2.X2 + 1
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2.X
[isNatKind](X) = 2.X + 2
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[tt] = 2
[U11#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U12#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U13#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U14#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U15#](X1,X2) = X1 + 2.X2 + 2
[U21#](X1,X2) = 2.X2 + 2
[U22#](X1,X2) = 2.X2 + 2
[U31#](X1,X2) = 2.X1 + 2.X2 + 1
[U51#](X1,X2) = 2.X2 + 2
[U52#](X1,X2) = 2.X2 + 1
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = 2.X + 1
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = 2.X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U12#(tt,V1:S,V2:S) -> U13#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U12#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U12#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U13#(tt,V1:S,V2:S) -> U14#(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U13#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U13#(tt,V1:S,V2:S) -> ISNATKIND(activate(V2:S))
 U14#(tt,V1:S,V2:S) -> U15#(isNat(activate(V1:S)),activate(V2:S))
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U14#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U14#(tt,V1:S,V2:S) -> ISNAT(activate(V1:S))
 U15#(tt,V2:S) -> ACTIVATE(V2:S)
 U15#(tt,V2:S) -> ISNAT(activate(V2:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V1:S)
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X1 + 2
[U12](X1,X2,X3) = X1 + 2
[U13](X1,X2,X3) = 2.X1 + 2
[U14](X1,X2,X3) = 2.X1 + 2
[U15](X1,X2) = 2
[U16](X) = 1
[U21](X1,X2) = 0
[U22](X1,X2) = 2.X1
[U23](X) = 0
[U31](X1,X2) = 2.X1
[U32](X) = 2.X
[U41](X) = 2.X
[U51](X1,X2) = 2.X1 + 2.X2 + 2
[U52](X1,X2) = 2.X1 + 2.X2 + 2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2
[isNatKind](X) = 0
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[tt] = 0
[U11#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U21#](X1,X2) = 2.X1 + 2.X2 + 1
[U22#](X1,X2) = 2.X1 + 2.X2 + 1
[U31#](X1,X2) = 2.X1 + 2.X2 + 2
[U51#](X1,X2) = X1 + 2.X2 + 1
[U52#](X1,X2) = 2.X2 + 1
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[ACTIVATE](X) = 2.X + 1
[ISNAT](X) = 2.X + 1
[ISNATKIND](X) = 2.X + 1
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U11#(tt,V1:S,V2:S) -> ACTIVATE(V2:S)
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1
[U12](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U13](X1,X2,X3) = X1 + 2.X2 + X3 + 2
[U14](X1,X2,X3) = X1 + 2.X2 + 2
[U15](X1,X2) = X1 + 2
[U16](X) = 2
[U21](X1,X2) = 2.X2 + 2
[U22](X1,X2) = 2.X1 + 2
[U23](X) = 2
[U31](X1,X2) = 2.X1 + 2.X2
[U32](X) = 2.X + 2
[U41](X) = 2
[U51](X1,X2) = 2.X2 + 2
[U52](X1,X2) = X1 + X2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2.X + 1
[isNatKind](X) = X
[plus](X1,X2) = 2.X1 + 2.X2
[s](X) = X + 2
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2
[n__s](X) = X + 2
[tt] = 2
[U11#](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 1
[U21#](X1,X2) = 2.X2 + 2
[U22#](X1,X2) = 2.X2 + 1
[U31#](X1,X2) = 2.X1 + 2.X2
[U51#](X1,X2) = 2.X2 + 2
[U52#](X1,X2) = X1 + X2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 1
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 1
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 1
[ACTIVATE](X) = X + 1
[ISNAT](X) = 2.X + 1
[ISNATKIND](X) = 2.X + 1
[PLUS](X1,X2) = 2.X1 + 2.X2 + 1

Problem 1: 

SCC Processor:
-> Pairs:
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U11#(tt,V1:S,V2:S) -> ISNATKIND(activate(V1:S))
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X1 + 2.X3 + 1
[U12](X1,X2,X3) = 2.X3 + 2
[U13](X1,X2,X3) = 2.X3 + 2
[U14](X1,X2,X3) = X1 + X3 + 2
[U15](X1,X2) = X2 + 2
[U16](X) = X + 1
[U21](X1,X2) = X1 + 1
[U22](X1,X2) = 2
[U23](X) = 2
[U31](X1,X2) = 2
[U32](X) = 2
[U41](X) = X
[U51](X1,X2) = X1 + X2
[U52](X1,X2) = X2 + 1
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = X + 1
[isNatKind](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[tt] = 2
[U11#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U21#](X1,X2) = 2.X2 + 1
[U22#](X1,X2) = 2.X2 + 1
[U31#](X1,X2) = 2.X2 + 2
[U51#](X1,X2) = 2.X2 + 2
[U52#](X1,X2) = 2.X2 + 2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = 2.X + 1
[ISNAT](X) = 2.X + 1
[ISNATKIND](X) = 2.X + 1
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> U11#(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U21#(tt,V1:S) -> U22#(isNatKind(activate(V1:S)),activate(V1:S))
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U12](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U13](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U14](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U15](X1,X2) = 2.X1 + 2.X2
[U16](X) = 2.X + 2
[U21](X1,X2) = X2 + 2
[U22](X1,X2) = X2 + 2
[U23](X) = X + 1
[U31](X1,X2) = 2
[U32](X) = X
[U41](X) = 2
[U51](X1,X2) = X1 + X2 + 1
[U52](X1,X2) = X1 + X2 + 1
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = X1 + 2.X2 + 2.X3
[activate](X) = X
[isNat](X) = X + 1
[isNatKind](X) = 2
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 1
[n__s](X) = X + 1
[tt] = 2
[U21#](X1,X2) = X1 + 2.X2 + 2
[U22#](X1,X2) = X1 + 2.X2 + 1
[U31#](X1,X2) = 2.X2 + 2
[U51#](X1,X2) = X1 + X2 + 2
[U52#](X1,X2) = X2 + 2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = X + 2
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U22#(tt,V1:S) -> ACTIVATE(V1:S)
 U22#(tt,V1:S) -> ISNAT(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U21#(tt,V1:S) -> ACTIVATE(V1:S)
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 0
[U11](X1,X2,X3) = 2.X2 + X3 + 2
[U12](X1,X2,X3) = X1 + 2.X2
[U13](X1,X2,X3) = X1 + 2.X2
[U14](X1,X2,X3) = 2.X2 + 2
[U15](X1,X2) = 2
[U16](X) = 2
[U21](X1,X2) = 2.X1 + 2.X2 + 2
[U22](X1,X2) = 2.X2 + 2
[U23](X) = 2
[U31](X1,X2) = X1
[U32](X) = 2
[U41](X) = 2
[U51](X1,X2) = 2.X2
[U52](X1,X2) = X2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = X1 + 2.X2 + 2.X3
[activate](X) = X
[isNat](X) = 2.X + 2
[isNatKind](X) = 2
[plus](X1,X2) = 2.X1 + 2.X2
[s](X) = X + 2
[n__0] = 0
[n__plus](X1,X2) = 2.X1 + 2.X2
[n__s](X) = X + 2
[tt] = 2
[U21#](X1,X2) = 2.X1 + 2.X2 + 1
[U31#](X1,X2) = 2.X2 + 2
[U51#](X1,X2) = 2.X2 + 2
[U52#](X1,X2) = X1 + 2.X2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = X1 + 2.X2 + 2.X3
[ACTIVATE](X) = 2.X + 2
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = 2.X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U21#(tt,V1:S) -> ISNATKIND(activate(V1:S))
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2
[U12](X1,X2,X3) = 2
[U13](X1,X2,X3) = 2
[U14](X1,X2,X3) = 2
[U15](X1,X2) = X1
[U16](X) = 2
[U21](X1,X2) = 2
[U22](X1,X2) = 2
[U23](X) = 2
[U31](X1,X2) = 2.X2 + 2
[U32](X) = 2
[U41](X) = 2
[U51](X1,X2) = 2.X2 + 2
[U52](X1,X2) = 2.X2 + 2
[U61](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2
[isNatKind](X) = X + 2
[plus](X1,X2) = 2.X1 + 2.X2
[s](X) = X + 2
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2
[n__s](X) = X + 2
[tt] = 2
[U21#](X1,X2) = X1 + X2 + 2
[U31#](X1,X2) = 2.X2 + 2
[U51#](X1,X2) = 2.X1 + 2.X2 + 2
[U52#](X1,X2) = X1 + X2 + 1
[U61#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = X + 2
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> U21#(isNatKind(activate(V1:S)),activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U31#(tt,V2:S) -> ACTIVATE(V2:S)
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X1 + 2.X3 + 2
[U12](X1,X2,X3) = 2.X3 + 2
[U13](X1,X2,X3) = X1 + X3 + 1
[U14](X1,X2,X3) = X1 + 2
[U15](X1,X2) = 2
[U16](X) = 2
[U21](X1,X2) = X1 + 1
[U22](X1,X2) = 2
[U23](X) = 2
[U31](X1,X2) = 2.X1 + 2.X2 + 1
[U32](X) = 2
[U41](X) = X
[U51](X1,X2) = X1 + X2 + 2
[U52](X1,X2) = X2 + 2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = X + 2
[isNatKind](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 1
[n__s](X) = X + 1
[tt] = 2
[U31#](X1,X2) = 2.X1 + 2.X2 + 2
[U51#](X1,X2) = X1 + X2
[U52#](X1,X2) = X2 + 2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = X + 2
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 1

Problem 1: 

SCC Processor:
-> Pairs:
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U31#(tt,V2:S) -> ISNATKIND(activate(V2:S))
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2
[U12](X1,X2,X3) = X1
[U13](X1,X2,X3) = 2
[U14](X1,X2,X3) = 2
[U15](X1,X2) = 2
[U16](X) = 2
[U21](X1,X2) = X1
[U22](X1,X2) = X1
[U23](X) = 2
[U31](X1,X2) = 2
[U32](X) = 2
[U41](X) = X
[U51](X1,X2) = 2.X1 + 2.X2 + 2
[U52](X1,X2) = X1 + 2.X2 + 1
[U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U64](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2
[isNatKind](X) = 2
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X + 2
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X + 2
[tt] = 2
[U31#](X1,X2) = X1 + 2.X2 + 2
[U51#](X1,X2) = 2.X1 + 2.X2 + 2
[U52#](X1,X2) = 2.X1 + 2.X2 + 2
[U61#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U62#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[ACTIVATE](X) = 2.X + 2
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = 2.X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> U31#(isNatKind(activate(V1:S)),activate(V2:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U51#(tt,N:S) -> U52#(isNatKind(activate(N:S)),activate(N:S))
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U12](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U13](X1,X2,X3) = X1 + 2.X2 + X3 + 1
[U14](X1,X2,X3) = X1 + 2.X2 + 2
[U15](X1,X2) = X1 + 1
[U16](X) = 2
[U21](X1,X2) = 2.X2 + 2
[U22](X1,X2) = 2.X1 + 2
[U23](X) = 2
[U31](X1,X2) = 2.X1 + 2.X2 + 1
[U32](X) = 2.X
[U41](X) = X + 1
[U51](X1,X2) = 2.X2 + 2
[U52](X1,X2) = X1 + X2 + 2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2.X + 2
[isNatKind](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 1
[n__s](X) = X + 1
[tt] = 2
[U51#](X1,X2) = 2.X2 + 2
[U52#](X1,X2) = 2.X2 + 1
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = 2.X
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = 2.X + 1
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U52#(tt,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U51#(tt,N:S) -> ACTIVATE(N:S)
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2
[U12](X1,X2,X3) = 2
[U13](X1,X2,X3) = 2
[U14](X1,X2,X3) = 2
[U15](X1,X2) = 2
[U16](X) = 2
[U21](X1,X2) = 2
[U22](X1,X2) = 2
[U23](X) = X
[U31](X1,X2) = 2.X1 + 1
[U32](X) = 2
[U41](X) = X + 2
[U51](X1,X2) = X2 + 1
[U52](X1,X2) = X2 + 1
[U61](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2
[isNatKind](X) = 2.X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 1
[n__s](X) = X + 1
[tt] = 2
[U51#](X1,X2) = 2.X1 + 2.X2 + 2
[U61#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = 2.X + 2
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = 2.X
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U51#(tt,N:S) -> ISNATKIND(activate(N:S))
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U12](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U13](X1,X2,X3) = X1 + 2.X2 + X3 + 1
[U14](X1,X2,X3) = X1 + 2.X2 + 2
[U15](X1,X2) = 2.X1 + 2
[U16](X) = 2
[U21](X1,X2) = X1
[U22](X1,X2) = 2
[U23](X) = 2
[U31](X1,X2) = X1 + 2.X2 + 2
[U32](X) = 2.X + 2
[U41](X) = X
[U51](X1,X2) = 2.X2 + 2
[U52](X1,X2) = X1 + X2 + 1
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = X + 1
[isNatKind](X) = X + 1
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[tt] = 2
[U51#](X1,X2) = 2.X2 + 2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = 2.X + 1
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = 2.X + 1
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> U51#(isNat(N:S),N:S)
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U61#(tt,M:S,N:S) -> U62#(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U13](X1,X2,X3) = X1 + 2.X2 + X3 + 2
[U14](X1,X2,X3) = X1 + 2.X2
[U15](X1,X2) = X1 + 2
[U16](X) = 2
[U21](X1,X2) = 2.X1 + 2
[U22](X1,X2) = 2
[U23](X) = 2
[U31](X1,X2) = 2.X1 + 2.X2 + 1
[U32](X) = 2.X + 1
[U41](X) = X + 1
[U51](X1,X2) = X2 + 2
[U52](X1,X2) = X2 + 2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2.X
[isNatKind](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 1
[n__s](X) = X + 1
[tt] = 2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62#](X1,X2,X3) = 2.X2 + 2.X3 + 1
[U63#](X1,X2,X3) = 2.X2 + 2.X3 + 1
[U64#](X1,X2,X3) = 2.X2 + 2.X3 + 1
[ACTIVATE](X) = 2.X + 1
[ISNAT](X) = 2.X + 1
[ISNATKIND](X) = 2.X + 1
[PLUS](X1,X2) = 2.X1 + 2.X2 + 1

Problem 1: 

SCC Processor:
-> Pairs:
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 U62#(tt,M:S,N:S) -> U63#(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U62#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U62#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U62#(tt,M:S,N:S) -> ISNAT(activate(N:S))
 U63#(tt,M:S,N:S) -> U64#(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U63#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U63#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U63#(tt,M:S,N:S) -> ISNATKIND(activate(N:S))
 U64#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U64#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U64#(tt,M:S,N:S) -> PLUS(activate(N:S),activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U61#(tt,M:S,N:S) -> ACTIVATE(M:S)
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2,X3) = 2.X1 + 2.X3 + 2
[U13](X1,X2,X3) = X1 + X3 + 1
[U14](X1,X2,X3) = X1 + 1
[U15](X1,X2) = 2
[U16](X) = 2
[U21](X1,X2) = 2.X2 + 2
[U22](X1,X2) = X1 + X2
[U23](X) = 2
[U31](X1,X2) = X1 + 2.X2 + 2
[U32](X) = 2.X
[U41](X) = 2
[U51](X1,X2) = 2.X2
[U52](X1,X2) = 2.X2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2.X + 2
[isNatKind](X) = X + 2
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[tt] = 2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = X + 1
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U61#(tt,M:S,N:S) -> ACTIVATE(N:S)
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 1
[U12](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U13](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U14](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U15](X1,X2) = X1 + 2.X2 + 1
[U16](X) = X + 2
[U21](X1,X2) = 2.X2 + 1
[U22](X1,X2) = 2.X2 + 1
[U23](X) = X
[U31](X1,X2) = 2.X1 + X2 + 1
[U32](X) = 2
[U41](X) = 2
[U51](X1,X2) = 2.X2 + 1
[U52](X1,X2) = 2.X2 + 1
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2.X + 1
[isNatKind](X) = 2.X + 2
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[tt] = 2
[U61#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[ACTIVATE](X) = 2.X + 1
[ISNAT](X) = 2.X + 1
[ISNATKIND](X) = 2.X + 1
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 U61#(tt,M:S,N:S) -> ISNATKIND(activate(M:S))
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X1 + 2.X3 + 1
[U12](X1,X2,X3) = 2.X3 + 2
[U13](X1,X2,X3) = X1 + X3 + 2
[U14](X1,X2,X3) = X1 + 1
[U15](X1,X2) = 2
[U16](X) = 2
[U21](X1,X2) = X2 + 1
[U22](X1,X2) = X1 + 1
[U23](X) = 2
[U31](X1,X2) = 2.X1 + 2.X2 + 2
[U32](X) = 2.X + 2
[U41](X) = X
[U51](X1,X2) = X1 + X2 + 2
[U52](X1,X2) = X2 + 2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = X + 2
[isNatKind](X) = X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[tt] = 2
[U61#](X1,X2,X3) = X1 + X2
[ACTIVATE](X) = X
[ISNAT](X) = 2.X + 1
[ISNATKIND](X) = X
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> U61#(isNat(M:S),M:S,N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X1:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 1
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U12](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U14](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U15](X1,X2) = X1 + 2.X2 + 2
[U16](X) = X + 1
[U21](X1,X2) = X1 + 2.X2 + 1
[U22](X1,X2) = 2.X2 + 2
[U23](X) = 2
[U31](X1,X2) = 2
[U32](X) = 2
[U41](X) = 2
[U51](X1,X2) = 2.X2
[U52](X1,X2) = X2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = X1 + 2.X2 + 2.X3
[activate](X) = X
[isNat](X) = 2.X + 2
[isNatKind](X) = 2
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[n__0] = 1
[n__plus](X1,X2) = 2.X1 + 2.X2 + 1
[n__s](X) = X + 1
[tt] = 2
[ACTIVATE](X) = 2.X + 2
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = 2.X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__plus(X1:S,X2:S)) -> ACTIVATE(X2:S)
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2
[U12](X1,X2,X3) = X1
[U13](X1,X2,X3) = 2
[U14](X1,X2,X3) = X1
[U15](X1,X2) = 2
[U16](X) = X
[U21](X1,X2) = X1
[U22](X1,X2) = X1
[U23](X) = 2
[U31](X1,X2) = 2
[U32](X) = 2
[U41](X) = X
[U51](X1,X2) = 2.X1 + X2
[U52](X1,X2) = X1 + X2 + 2
[U61](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2
[isNatKind](X) = 2
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X + 2
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X + 2
[tt] = 2
[ACTIVATE](X) = 2.X + 1
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = 2.X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 1

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1: 

Reduction Pair Processor:
-> Pairs:
 ACTIVATE(n__plus(X1:S,X2:S)) -> PLUS(activate(X1:S),activate(X2:S))
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X1 + 2.X3 + 2
[U12](X1,X2,X3) = 2.X3
[U13](X1,X2,X3) = X1
[U14](X1,X2,X3) = 2
[U15](X1,X2) = 2
[U16](X) = 2
[U21](X1,X2) = 2.X2 + 2
[U22](X1,X2) = X2 + 2
[U23](X) = 2
[U31](X1,X2) = 2.X1 + 2.X2 + 2
[U32](X) = 2
[U41](X) = X
[U51](X1,X2) = 2.X2 + 2
[U52](X1,X2) = 2.X2 + 2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2.X + 2
[isNatKind](X) = 2.X
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[tt] = 2
[ACTIVATE](X) = 2.X + 2
[ISNAT](X) = 2.X + 2
[ISNATKIND](X) = 2.X + 2
[PLUS](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1: 

SCC Processor:
-> Pairs:
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNAT(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNAT(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNAT(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ACTIVATE(V2:S)
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ACTIVATE(V1:S)
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
 PLUS(N:S,0) -> ISNAT(N:S)
 PLUS(N:S,s(M:S)) -> ISNAT(M:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->->Cycle:
->->-> Pairs:
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 ACTIVATE(n__s(X:S)) -> ACTIVATE(X:S)
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Projection:
 pi(ACTIVATE) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 ISNATKIND(n__plus(V1:S,V2:S)) -> ISNATKIND(activate(V1:S))
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 1
[U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2,X3) = X1 + X2 + 2.X3 + 2
[U13](X1,X2,X3) = X1 + X2 + X3 + 2
[U14](X1,X2,X3) = X1 + X2 + 1
[U15](X1,X2) = 2
[U16](X) = 2
[U21](X1,X2) = 2.X2
[U22](X1,X2) = 2.X2
[U23](X) = X
[U31](X1,X2) = X1 + 2.X2 + 2
[U32](X) = X + 2
[U41](X) = 2
[U51](X1,X2) = 2.X2 + 2
[U52](X1,X2) = 2.X2 + 2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = 2.X2 + 2.X3 + 2
[activate](X) = X
[isNat](X) = 2.X
[isNatKind](X) = X + 2
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[s](X) = X
[n__0] = 1
[n__plus](X1,X2) = 2.X1 + 2.X2 + 2
[n__s](X) = X
[tt] = 2
[ISNATKIND](X) = X

Problem 1.2: 

SCC Processor:
-> Pairs:
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
->->-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 ISNATKIND(n__s(V1:S)) -> ISNATKIND(activate(V1:S))
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
-> Usable rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[0] = 2
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U12](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U13](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U14](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U15](X1,X2) = X1 + 2.X2
[U16](X) = X
[U21](X1,X2) = 2.X1 + 2.X2
[U22](X1,X2) = X1 + 2.X2 + 2
[U23](X) = 2
[U31](X1,X2) = X1
[U32](X) = X
[U41](X) = 2
[U51](X1,X2) = 2.X2 + 2
[U52](X1,X2) = X1 + X2
[U61](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U62](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U63](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U64](X1,X2,X3) = X1 + 2.X2 + 2.X3
[activate](X) = X
[isNat](X) = 2.X + 2
[isNatKind](X) = 2
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[s](X) = X + 1
[n__0] = 2
[n__plus](X1,X2) = 2.X1 + 2.X2 + 1
[n__s](X) = X + 1
[tt] = 2
[ISNATKIND](X) = 2.X

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 0 -> n__0
 U11(tt,V1:S,V2:S) -> U12(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 U12(tt,V1:S,V2:S) -> U13(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U13(tt,V1:S,V2:S) -> U14(isNatKind(activate(V2:S)),activate(V1:S),activate(V2:S))
 U14(tt,V1:S,V2:S) -> U15(isNat(activate(V1:S)),activate(V2:S))
 U15(tt,V2:S) -> U16(isNat(activate(V2:S)))
 U16(tt) -> tt
 U21(tt,V1:S) -> U22(isNatKind(activate(V1:S)),activate(V1:S))
 U22(tt,V1:S) -> U23(isNat(activate(V1:S)))
 U23(tt) -> tt
 U31(tt,V2:S) -> U32(isNatKind(activate(V2:S)))
 U32(tt) -> tt
 U41(tt) -> tt
 U51(tt,N:S) -> U52(isNatKind(activate(N:S)),activate(N:S))
 U52(tt,N:S) -> activate(N:S)
 U61(tt,M:S,N:S) -> U62(isNatKind(activate(M:S)),activate(M:S),activate(N:S))
 U62(tt,M:S,N:S) -> U63(isNat(activate(N:S)),activate(M:S),activate(N:S))
 U63(tt,M:S,N:S) -> U64(isNatKind(activate(N:S)),activate(M:S),activate(N:S))
 U64(tt,M:S,N:S) -> s(plus(activate(N:S),activate(M:S)))
 activate(n__0) -> 0
 activate(n__plus(X1:S,X2:S)) -> plus(activate(X1:S),activate(X2:S))
 activate(n__s(X:S)) -> s(activate(X:S))
 activate(X:S) -> X:S
 isNat(n__0) -> tt
 isNat(n__plus(V1:S,V2:S)) -> U11(isNatKind(activate(V1:S)),activate(V1:S),activate(V2:S))
 isNat(n__s(V1:S)) -> U21(isNatKind(activate(V1:S)),activate(V1:S))
 isNatKind(n__0) -> tt
 isNatKind(n__plus(V1:S,V2:S)) -> U31(isNatKind(activate(V1:S)),activate(V2:S))
 isNatKind(n__s(V1:S)) -> U41(isNatKind(activate(V1:S)))
 plus(N:S,0) -> U51(isNat(N:S),N:S)
 plus(N:S,s(M:S)) -> U61(isNat(M:S),M:S,N:S)
 plus(X1:S,X2:S) -> n__plus(X1:S,X2:S)
 s(X:S) -> n__s(X:S)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.