/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR M N V1 V2)
(STRATEGY CONTEXTSENSITIVE
(U101 1)
(U102 1)
(U103 1)
(U104 1)
(U11 1)
(U12 1)
(U13 1)
(U14 1)
(U15 1)
(U16 1)
(U21 1)
(U22 1)
(U23 1)
(U31 1)
(U32 1)
(U33 1)
(U34 1)
(U35 1)
(U36 1)
(U41 1)
(U42 1)
(U51 1)
(U61 1)
(U62 1)
(U71 1)
(U72 1)
(U81 1)
(U82 1)
(U83 1)
(U84 1)
(U91 1)
(U92 1)
(isNat)
(isNatKind)
(plus 1 2)
(x 1 2)
(0)
(s 1)
(tt)
)
(RULES
U101(tt,M,N) -> U102(isNatKind(M),M,N)
U102(tt,M,N) -> U103(isNat(N),M,N)
U103(tt,M,N) -> U104(isNatKind(N),M,N)
U104(tt,M,N) -> plus(x(N,M),N)
U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
U14(tt,V1,V2) -> U15(isNat(V1),V2)
U15(tt,V2) -> U16(isNat(V2))
U16(tt) -> tt
U21(tt,V1) -> U22(isNatKind(V1),V1)
U22(tt,V1) -> U23(isNat(V1))
U23(tt) -> tt
U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
U34(tt,V1,V2) -> U35(isNat(V1),V2)
U35(tt,V2) -> U36(isNat(V2))
U36(tt) -> tt
U41(tt,V2) -> U42(isNatKind(V2))
U42(tt) -> tt
U51(tt) -> tt
U61(tt,V2) -> U62(isNatKind(V2))
U62(tt) -> tt
U71(tt,N) -> U72(isNatKind(N),N)
U72(tt,N) -> N
U81(tt,M,N) -> U82(isNatKind(M),M,N)
U82(tt,M,N) -> U83(isNat(N),M,N)
U83(tt,M,N) -> U84(isNatKind(N),M,N)
U84(tt,M,N) -> s(plus(N,M))
U91(tt,N) -> U92(isNatKind(N))
U92(tt) -> 0
isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
isNat(0) -> tt
isNat(s(V1)) -> U21(isNatKind(V1),V1)
isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
isNatKind(0) -> tt
isNatKind(s(V1)) -> U51(isNatKind(V1))
plus(N,0) -> U71(isNat(N),N)
plus(N,s(M)) -> U81(isNat(M),M,N)
x(N,0) -> U91(isNat(N),N)
x(N,s(M)) -> U101(isNat(M),M,N)
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 U101#(tt,M,N) -> U102#(isNatKind(M),M,N)
 U101#(tt,M,N) -> ISNATKIND(M)
 U102#(tt,M,N) -> U103#(isNat(N),M,N)
 U102#(tt,M,N) -> ISNAT(N)
 U103#(tt,M,N) -> U104#(isNatKind(N),M,N)
 U103#(tt,M,N) -> ISNATKIND(N)
 U104#(tt,M,N) -> PLUS(x(N,M),N)
 U104#(tt,M,N) -> X(N,M)
 U104#(tt,M,N) -> M
 U104#(tt,M,N) -> N
 U11#(tt,V1,V2) -> U12#(isNatKind(V1),V1,V2)
 U11#(tt,V1,V2) -> ISNATKIND(V1)
 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2)
 U12#(tt,V1,V2) -> ISNATKIND(V2)
 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2)
 U13#(tt,V1,V2) -> ISNATKIND(V2)
 U14#(tt,V1,V2) -> U15#(isNat(V1),V2)
 U14#(tt,V1,V2) -> ISNAT(V1)
 U15#(tt,V2) -> U16#(isNat(V2))
 U15#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> U22#(isNatKind(V1),V1)
 U21#(tt,V1) -> ISNATKIND(V1)
 U22#(tt,V1) -> U23#(isNat(V1))
 U22#(tt,V1) -> ISNAT(V1)
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U31#(tt,V1,V2) -> ISNATKIND(V1)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U32#(tt,V1,V2) -> ISNATKIND(V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> ISNATKIND(V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> U36#(isNat(V2))
 U35#(tt,V2) -> ISNAT(V2)
 U41#(tt,V2) -> U42#(isNatKind(V2))
 U41#(tt,V2) -> ISNATKIND(V2)
 U61#(tt,V2) -> U62#(isNatKind(V2))
 U61#(tt,V2) -> ISNATKIND(V2)
 U71#(tt,N) -> U72#(isNatKind(N),N)
 U71#(tt,N) -> ISNATKIND(N)
 U72#(tt,N) -> N
 U81#(tt,M,N) -> U82#(isNatKind(M),M,N)
 U81#(tt,M,N) -> ISNATKIND(M)
 U82#(tt,M,N) -> U83#(isNat(N),M,N)
 U82#(tt,M,N) -> ISNAT(N)
 U83#(tt,M,N) -> U84#(isNatKind(N),M,N)
 U83#(tt,M,N) -> ISNATKIND(N)
 U84#(tt,M,N) -> PLUS(N,M)
 U84#(tt,M,N) -> M
 U84#(tt,M,N) -> N
 U91#(tt,N) -> U92#(isNatKind(N))
 U91#(tt,N) -> ISNATKIND(N)
 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2)
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> U41#(isNatKind(V1),V2)
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(x(V1,V2)) -> U61#(isNatKind(V1),V2)
 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> U51#(isNatKind(V1))
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
 PLUS(N,0) -> U71#(isNat(N),N)
 PLUS(N,0) -> ISNAT(N)
 PLUS(N,s(M)) -> U81#(isNat(M),M,N)
 PLUS(N,s(M)) -> ISNAT(M)
 X(N,0) -> U91#(isNat(N),N)
 X(N,0) -> ISNAT(N)
 X(N,s(M)) -> U101#(isNat(M),M,N)
 X(N,s(M)) -> ISNAT(M)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding Rules:
 Empty

Problem 1: 

SCC Processor:
-> Pairs:
 U101#(tt,M,N) -> U102#(isNatKind(M),M,N)
 U101#(tt,M,N) -> ISNATKIND(M)
 U102#(tt,M,N) -> U103#(isNat(N),M,N)
 U102#(tt,M,N) -> ISNAT(N)
 U103#(tt,M,N) -> U104#(isNatKind(N),M,N)
 U103#(tt,M,N) -> ISNATKIND(N)
 U104#(tt,M,N) -> PLUS(x(N,M),N)
 U104#(tt,M,N) -> X(N,M)
 U104#(tt,M,N) -> M
 U104#(tt,M,N) -> N
 U11#(tt,V1,V2) -> U12#(isNatKind(V1),V1,V2)
 U11#(tt,V1,V2) -> ISNATKIND(V1)
 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2)
 U12#(tt,V1,V2) -> ISNATKIND(V2)
 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2)
 U13#(tt,V1,V2) -> ISNATKIND(V2)
 U14#(tt,V1,V2) -> U15#(isNat(V1),V2)
 U14#(tt,V1,V2) -> ISNAT(V1)
 U15#(tt,V2) -> U16#(isNat(V2))
 U15#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> U22#(isNatKind(V1),V1)
 U21#(tt,V1) -> ISNATKIND(V1)
 U22#(tt,V1) -> U23#(isNat(V1))
 U22#(tt,V1) -> ISNAT(V1)
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U31#(tt,V1,V2) -> ISNATKIND(V1)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U32#(tt,V1,V2) -> ISNATKIND(V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> ISNATKIND(V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> U36#(isNat(V2))
 U35#(tt,V2) -> ISNAT(V2)
 U41#(tt,V2) -> U42#(isNatKind(V2))
 U41#(tt,V2) -> ISNATKIND(V2)
 U61#(tt,V2) -> U62#(isNatKind(V2))
 U61#(tt,V2) -> ISNATKIND(V2)
 U71#(tt,N) -> U72#(isNatKind(N),N)
 U71#(tt,N) -> ISNATKIND(N)
 U72#(tt,N) -> N
 U81#(tt,M,N) -> U82#(isNatKind(M),M,N)
 U81#(tt,M,N) -> ISNATKIND(M)
 U82#(tt,M,N) -> U83#(isNat(N),M,N)
 U82#(tt,M,N) -> ISNAT(N)
 U83#(tt,M,N) -> U84#(isNatKind(N),M,N)
 U83#(tt,M,N) -> ISNATKIND(N)
 U84#(tt,M,N) -> PLUS(N,M)
 U84#(tt,M,N) -> M
 U84#(tt,M,N) -> N
 U91#(tt,N) -> U92#(isNatKind(N))
 U91#(tt,N) -> ISNATKIND(N)
 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2)
 ISNAT(plus(V1,V2)) -> ISNATKIND(V1)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
 ISNAT(x(V1,V2)) -> ISNATKIND(V1)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
 ISNAT(s(V1)) -> ISNATKIND(V1)
 ISNATKIND(plus(V1,V2)) -> U41#(isNatKind(V1),V2)
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(x(V1,V2)) -> U61#(isNatKind(V1),V2)
 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> U51#(isNatKind(V1))
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
 PLUS(N,0) -> U71#(isNat(N),N)
 PLUS(N,0) -> ISNAT(N)
 PLUS(N,s(M)) -> U81#(isNat(M),M,N)
 PLUS(N,s(M)) -> ISNAT(M)
 X(N,0) -> U91#(isNat(N),N)
 X(N,0) -> ISNAT(N)
 X(N,s(M)) -> U101#(isNat(M),M,N)
 X(N,s(M)) -> ISNAT(M)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U41#(tt,V2) -> ISNATKIND(V2)
 U61#(tt,V2) -> ISNATKIND(V2)
 ISNATKIND(plus(V1,V2)) -> U41#(isNatKind(V1),V2)
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(x(V1,V2)) -> U61#(isNatKind(V1),V2)
 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
->->-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
->->-> Unhiding rules:
 Empty
->->Cycle:
->->-> Pairs:
 U11#(tt,V1,V2) -> U12#(isNatKind(V1),V1,V2)
 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2)
 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2)
 U14#(tt,V1,V2) -> U15#(isNat(V1),V2)
 U14#(tt,V1,V2) -> ISNAT(V1)
 U15#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> U22#(isNatKind(V1),V1)
 U22#(tt,V1) -> ISNAT(V1)
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> ISNAT(V2)
 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
->->-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
->->-> Unhiding rules:
 Empty
->->Cycle:
->->-> Pairs:
 U81#(tt,M,N) -> U82#(isNatKind(M),M,N)
 U82#(tt,M,N) -> U83#(isNat(N),M,N)
 U83#(tt,M,N) -> U84#(isNatKind(N),M,N)
 U84#(tt,M,N) -> PLUS(N,M)
 PLUS(N,s(M)) -> U81#(isNat(M),M,N)
->->-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
->->-> Unhiding rules:
 Empty
->->Cycle:
->->-> Pairs:
 U101#(tt,M,N) -> U102#(isNatKind(M),M,N)
 U102#(tt,M,N) -> U103#(isNat(N),M,N)
 U103#(tt,M,N) -> U104#(isNatKind(N),M,N)
 U104#(tt,M,N) -> X(N,M)
 X(N,s(M)) -> U101#(isNat(M),M,N)
->->-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
->->-> Unhiding rules:
 Empty


The problem is decomposed in 4 subproblems.

Problem 1.1: 

SubNColl Processor:
-> Pairs:
 U41#(tt,V2) -> ISNATKIND(V2)
 U61#(tt,V2) -> ISNATKIND(V2)
 ISNATKIND(plus(V1,V2)) -> U41#(isNatKind(V1),V2)
 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(x(V1,V2)) -> U61#(isNatKind(V1),V2)
 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1)
 ISNATKIND(s(V1)) -> ISNATKIND(V1)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Projection:
 pi(U41#) = 2
 pi(U61#) = 2
 pi(ISNATKIND) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 U41#(tt,V2) -> ISNATKIND(V2)
 U61#(tt,V2) -> ISNATKIND(V2)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 U11#(tt,V1,V2) -> U12#(isNatKind(V1),V1,V2)
 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2)
 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2)
 U14#(tt,V1,V2) -> U15#(isNat(V1),V2)
 U14#(tt,V1,V2) -> ISNAT(V1)
 U15#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> U22#(isNatKind(V1),V1)
 U22#(tt,V1) -> ISNAT(V1)
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> ISNAT(V2)
 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
-> Usable rules:
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2
[U13](X1,X2,X3) = X1 + 2.X2 + X3 + 2
[U14](X1,X2,X3) = 2.X1 + 2.X2 + X3
[U15](X1,X2) = 2.X1 + X2
[U16](X) = X + 2
[U21](X1,X2) = X1 + 2.X2 + 2
[U22](X1,X2) = X1 + 2.X2 + 2
[U23](X) = X + 2
[U31](X1,X2,X3) = X2 + 2.X3 + 2
[U32](X1,X2,X3) = X1 + X2 + 2.X3
[U33](X1,X2,X3) = X1 + X2 + 2.X3
[U34](X1,X2,X3) = X2 + X3 + 2
[U35](X1,X2) = X2 + 2
[U36](X) = X
[U41](X1,X2) = X1
[U42](X) = 2
[U51](X) = 2
[U61](X1,X2) = 2
[U62](X) = X
[isNat](X) = X + 2
[isNatKind](X) = 2
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[x](X1,X2) = 2.X1 + 2.X2
[0] = 0
[s](X) = 2.X + 2
[tt] = 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 + 2.X3
[U14#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U15#](X1,X2) = 2.X2 + 2
[U21#](X1,X2) = 2.X2 + 2
[U22#](X1,X2) = 2.X2 + 2
[U31#](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U32#](X1,X2,X3) = X1 + 2.X2 + 2.X3
[U33#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U34#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U35#](X1,X2) = X1 + 2.X2
[ISNAT](X) = 2.X + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2)
 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2)
 U14#(tt,V1,V2) -> U15#(isNat(V1),V2)
 U14#(tt,V1,V2) -> ISNAT(V1)
 U15#(tt,V2) -> ISNAT(V2)
 U21#(tt,V1) -> U22#(isNatKind(V1),V1)
 U22#(tt,V1) -> ISNAT(V1)
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> ISNAT(V2)
 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U21#(tt,V1) -> U22#(isNatKind(V1),V1)
 U22#(tt,V1) -> ISNAT(V1)
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> ISNAT(V2)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
->->-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
->->-> Unhiding rules:
 Empty

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 U21#(tt,V1) -> U22#(isNatKind(V1),V1)
 U22#(tt,V1) -> ISNAT(V1)
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> ISNAT(V2)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
-> Usable rules:
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U12](X1,X2,X3) = X1 + 2.X3 + 1
[U13](X1,X2,X3) = 2.X3 + 2
[U14](X1,X2,X3) = X1 + 2
[U15](X1,X2) = 2
[U16](X) = 2
[U21](X1,X2) = X1 + 2.X2 + 2
[U22](X1,X2) = X1 + 2
[U23](X) = 2
[U31](X1,X2,X3) = 2.X1 + 2.X3 + 2
[U32](X1,X2,X3) = 2.X3 + 2
[U33](X1,X2,X3) = 2.X3 + 2
[U34](X1,X2,X3) = X1 + 2
[U35](X1,X2) = 2
[U36](X) = 2
[U41](X1,X2) = 2.X1 + 2.X2 + 2
[U42](X) = X
[U51](X) = 2.X
[U61](X1,X2) = 2.X1 + 2.X2
[U62](X) = X + 2
[isNat](X) = 2.X
[isNatKind](X) = 2.X
[plus](X1,X2) = 2.X1 + 2.X2 + 1
[x](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 2
[s](X) = 2.X + 2
[tt] = 2
[U21#](X1,X2) = X1 + 2.X2 + 2
[U22#](X1,X2) = 2.X2 + 2
[U31#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U32#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U33#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U34#](X1,X2,X3) = 2.X2 + 2.X3 + 2
[U35#](X1,X2) = X1 + 2.X2 + 2
[ISNAT](X) = 2.X + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 U22#(tt,V1) -> ISNAT(V1)
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> ISNAT(V2)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> ISNAT(V2)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
->->-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
->->-> Unhiding rules:
 Empty

Problem 1.2: 

Reduction Pairs Processor:
-> Pairs:
 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2)
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> ISNAT(V2)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
-> Usable rules:
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1
[U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U14](X1,X2,X3) = X1 + 2.X2 + 2
[U15](X1,X2) = X1
[U16](X) = 2
[U21](X1,X2) = X1
[U22](X1,X2) = 2
[U23](X) = 2
[U31](X1,X2,X3) = 2.X1 + 2.X3 + 2
[U32](X1,X2,X3) = 2.X1 + 2.X3 + 2
[U33](X1,X2,X3) = 2.X1 + 2.X3 + 2
[U34](X1,X2,X3) = X1 + 2.X3 + 2
[U35](X1,X2) = 2.X2 + 2
[U36](X) = X
[U41](X1,X2) = 2
[U42](X) = X
[U51](X) = X
[U61](X1,X2) = 2
[U62](X) = 2
[isNat](X) = 2.X + 2
[isNatKind](X) = 2
[plus](X1,X2) = 2.X1 + 2.X2 + 2
[x](X1,X2) = 2.X1 + 2.X2 + 2
[0] = 0
[s](X) = 2
[tt] = 2
[U31#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1
[U32#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U33#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3
[U34#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2
[U35#](X1,X2) = X1 + 2.X2 + 2
[ISNAT](X) = 2.X + 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2)
 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2)
 U34#(tt,V1,V2) -> U35#(isNat(V1),V2)
 U34#(tt,V1,V2) -> ISNAT(V1)
 U35#(tt,V2) -> ISNAT(V2)
 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

SubNColl Processor:
-> Pairs:
 U81#(tt,M,N) -> U82#(isNatKind(M),M,N)
 U82#(tt,M,N) -> U83#(isNat(N),M,N)
 U83#(tt,M,N) -> U84#(isNatKind(N),M,N)
 U84#(tt,M,N) -> PLUS(N,M)
 PLUS(N,s(M)) -> U81#(isNat(M),M,N)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Projection:
 pi(U81#) = 2
 pi(U82#) = 2
 pi(U83#) = 2
 pi(U84#) = 2
 pi(PLUS) = 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 U81#(tt,M,N) -> U82#(isNatKind(M),M,N)
 U82#(tt,M,N) -> U83#(isNat(N),M,N)
 U83#(tt,M,N) -> U84#(isNatKind(N),M,N)
 U84#(tt,M,N) -> PLUS(N,M)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

SubNColl Processor:
-> Pairs:
 U101#(tt,M,N) -> U102#(isNatKind(M),M,N)
 U102#(tt,M,N) -> U103#(isNat(N),M,N)
 U103#(tt,M,N) -> U104#(isNatKind(N),M,N)
 U104#(tt,M,N) -> X(N,M)
 X(N,s(M)) -> U101#(isNat(M),M,N)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Projection:
 pi(U101#) = 2
 pi(U102#) = 2
 pi(U103#) = 2
 pi(U104#) = 2
 pi(X) = 2

Problem 1.4: 

SCC Processor:
-> Pairs:
 U101#(tt,M,N) -> U102#(isNatKind(M),M,N)
 U102#(tt,M,N) -> U103#(isNat(N),M,N)
 U103#(tt,M,N) -> U104#(isNatKind(N),M,N)
 U104#(tt,M,N) -> X(N,M)
-> Rules:
 U101(tt,M,N) -> U102(isNatKind(M),M,N)
 U102(tt,M,N) -> U103(isNat(N),M,N)
 U103(tt,M,N) -> U104(isNatKind(N),M,N)
 U104(tt,M,N) -> plus(x(N,M),N)
 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2)
 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2)
 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2)
 U14(tt,V1,V2) -> U15(isNat(V1),V2)
 U15(tt,V2) -> U16(isNat(V2))
 U16(tt) -> tt
 U21(tt,V1) -> U22(isNatKind(V1),V1)
 U22(tt,V1) -> U23(isNat(V1))
 U23(tt) -> tt
 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2)
 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2)
 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2)
 U34(tt,V1,V2) -> U35(isNat(V1),V2)
 U35(tt,V2) -> U36(isNat(V2))
 U36(tt) -> tt
 U41(tt,V2) -> U42(isNatKind(V2))
 U42(tt) -> tt
 U51(tt) -> tt
 U61(tt,V2) -> U62(isNatKind(V2))
 U62(tt) -> tt
 U71(tt,N) -> U72(isNatKind(N),N)
 U72(tt,N) -> N
 U81(tt,M,N) -> U82(isNatKind(M),M,N)
 U82(tt,M,N) -> U83(isNat(N),M,N)
 U83(tt,M,N) -> U84(isNatKind(N),M,N)
 U84(tt,M,N) -> s(plus(N,M))
 U91(tt,N) -> U92(isNatKind(N))
 U92(tt) -> 0
 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2)
 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2)
 isNat(0) -> tt
 isNat(s(V1)) -> U21(isNatKind(V1),V1)
 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2)
 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2)
 isNatKind(0) -> tt
 isNatKind(s(V1)) -> U51(isNatKind(V1))
 plus(N,0) -> U71(isNat(N),N)
 plus(N,s(M)) -> U81(isNat(M),M,N)
 x(N,0) -> U91(isNat(N),N)
 x(N,s(M)) -> U101(isNat(M),M,N)
-> Unhiding rules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.