0.00/0.39 YES 0.00/0.39 0.00/0.39 Problem 1: 0.00/0.39 0.00/0.39 (VAR M N V1 V2) 0.00/0.39 (STRATEGY CONTEXTSENSITIVE 0.00/0.39 (U101 1) 0.00/0.39 (U102 1) 0.00/0.39 (U103 1) 0.00/0.39 (U104 1) 0.00/0.39 (U11 1) 0.00/0.39 (U12 1) 0.00/0.39 (U13 1) 0.00/0.39 (U14 1) 0.00/0.39 (U15 1) 0.00/0.39 (U16 1) 0.00/0.39 (U21 1) 0.00/0.39 (U22 1) 0.00/0.39 (U23 1) 0.00/0.39 (U31 1) 0.00/0.39 (U32 1) 0.00/0.39 (U33 1) 0.00/0.39 (U34 1) 0.00/0.39 (U35 1) 0.00/0.39 (U36 1) 0.00/0.39 (U41 1) 0.00/0.39 (U42 1) 0.00/0.39 (U51 1) 0.00/0.39 (U61 1) 0.00/0.39 (U62 1) 0.00/0.39 (U71 1) 0.00/0.39 (U72 1) 0.00/0.39 (U81 1) 0.00/0.39 (U82 1) 0.00/0.39 (U83 1) 0.00/0.39 (U84 1) 0.00/0.39 (U91 1) 0.00/0.39 (U92 1) 0.00/0.39 (isNat) 0.00/0.39 (isNatKind) 0.00/0.39 (plus 1 2) 0.00/0.39 (x 1 2) 0.00/0.39 (0) 0.00/0.39 (s 1) 0.00/0.39 (tt) 0.00/0.39 ) 0.00/0.39 (RULES 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 ) 0.00/0.39 0.00/0.39 Problem 1: 0.00/0.39 0.00/0.39 Dependency Pairs Processor: 0.00/0.39 -> Pairs: 0.00/0.39 U101#(tt,M,N) -> U102#(isNatKind(M),M,N) 0.00/0.39 U101#(tt,M,N) -> ISNATKIND(M) 0.00/0.39 U102#(tt,M,N) -> U103#(isNat(N),M,N) 0.00/0.39 U102#(tt,M,N) -> ISNAT(N) 0.00/0.39 U103#(tt,M,N) -> U104#(isNatKind(N),M,N) 0.00/0.39 U103#(tt,M,N) -> ISNATKIND(N) 0.00/0.39 U104#(tt,M,N) -> PLUS(x(N,M),N) 0.00/0.39 U104#(tt,M,N) -> X(N,M) 0.00/0.39 U104#(tt,M,N) -> M 0.00/0.39 U104#(tt,M,N) -> N 0.00/0.39 U11#(tt,V1,V2) -> U12#(isNatKind(V1),V1,V2) 0.00/0.39 U11#(tt,V1,V2) -> ISNATKIND(V1) 0.00/0.39 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2) 0.00/0.39 U12#(tt,V1,V2) -> ISNATKIND(V2) 0.00/0.39 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2) 0.00/0.39 U13#(tt,V1,V2) -> ISNATKIND(V2) 0.00/0.39 U14#(tt,V1,V2) -> U15#(isNat(V1),V2) 0.00/0.39 U14#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U15#(tt,V2) -> U16#(isNat(V2)) 0.00/0.39 U15#(tt,V2) -> ISNAT(V2) 0.00/0.39 U21#(tt,V1) -> U22#(isNatKind(V1),V1) 0.00/0.39 U21#(tt,V1) -> ISNATKIND(V1) 0.00/0.39 U22#(tt,V1) -> U23#(isNat(V1)) 0.00/0.39 U22#(tt,V1) -> ISNAT(V1) 0.00/0.39 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.39 U31#(tt,V1,V2) -> ISNATKIND(V1) 0.00/0.39 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.39 U32#(tt,V1,V2) -> ISNATKIND(V2) 0.00/0.39 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.39 U33#(tt,V1,V2) -> ISNATKIND(V2) 0.00/0.39 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.39 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U35#(tt,V2) -> U36#(isNat(V2)) 0.00/0.39 U35#(tt,V2) -> ISNAT(V2) 0.00/0.39 U41#(tt,V2) -> U42#(isNatKind(V2)) 0.00/0.39 U41#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 U61#(tt,V2) -> U62#(isNatKind(V2)) 0.00/0.39 U61#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 U71#(tt,N) -> U72#(isNatKind(N),N) 0.00/0.39 U71#(tt,N) -> ISNATKIND(N) 0.00/0.39 U72#(tt,N) -> N 0.00/0.39 U81#(tt,M,N) -> U82#(isNatKind(M),M,N) 0.00/0.39 U81#(tt,M,N) -> ISNATKIND(M) 0.00/0.39 U82#(tt,M,N) -> U83#(isNat(N),M,N) 0.00/0.39 U82#(tt,M,N) -> ISNAT(N) 0.00/0.39 U83#(tt,M,N) -> U84#(isNatKind(N),M,N) 0.00/0.39 U83#(tt,M,N) -> ISNATKIND(N) 0.00/0.39 U84#(tt,M,N) -> PLUS(N,M) 0.00/0.39 U84#(tt,M,N) -> M 0.00/0.39 U84#(tt,M,N) -> N 0.00/0.39 U91#(tt,N) -> U92#(isNatKind(N)) 0.00/0.39 U91#(tt,N) -> ISNATKIND(N) 0.00/0.39 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(plus(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(x(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1) 0.00/0.39 ISNAT(s(V1)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(plus(V1,V2)) -> U41#(isNatKind(V1),V2) 0.00/0.39 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(x(V1,V2)) -> U61#(isNatKind(V1),V2) 0.00/0.39 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(s(V1)) -> U51#(isNatKind(V1)) 0.00/0.39 ISNATKIND(s(V1)) -> ISNATKIND(V1) 0.00/0.39 PLUS(N,0) -> U71#(isNat(N),N) 0.00/0.39 PLUS(N,0) -> ISNAT(N) 0.00/0.39 PLUS(N,s(M)) -> U81#(isNat(M),M,N) 0.00/0.39 PLUS(N,s(M)) -> ISNAT(M) 0.00/0.39 X(N,0) -> U91#(isNat(N),N) 0.00/0.39 X(N,0) -> ISNAT(N) 0.00/0.39 X(N,s(M)) -> U101#(isNat(M),M,N) 0.00/0.39 X(N,s(M)) -> ISNAT(M) 0.00/0.39 -> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 -> Unhiding Rules: 0.00/0.39 Empty 0.00/0.39 0.00/0.39 Problem 1: 0.00/0.39 0.00/0.39 SCC Processor: 0.00/0.39 -> Pairs: 0.00/0.39 U101#(tt,M,N) -> U102#(isNatKind(M),M,N) 0.00/0.39 U101#(tt,M,N) -> ISNATKIND(M) 0.00/0.39 U102#(tt,M,N) -> U103#(isNat(N),M,N) 0.00/0.39 U102#(tt,M,N) -> ISNAT(N) 0.00/0.39 U103#(tt,M,N) -> U104#(isNatKind(N),M,N) 0.00/0.39 U103#(tt,M,N) -> ISNATKIND(N) 0.00/0.39 U104#(tt,M,N) -> PLUS(x(N,M),N) 0.00/0.39 U104#(tt,M,N) -> X(N,M) 0.00/0.39 U104#(tt,M,N) -> M 0.00/0.39 U104#(tt,M,N) -> N 0.00/0.39 U11#(tt,V1,V2) -> U12#(isNatKind(V1),V1,V2) 0.00/0.39 U11#(tt,V1,V2) -> ISNATKIND(V1) 0.00/0.39 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2) 0.00/0.39 U12#(tt,V1,V2) -> ISNATKIND(V2) 0.00/0.39 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2) 0.00/0.39 U13#(tt,V1,V2) -> ISNATKIND(V2) 0.00/0.39 U14#(tt,V1,V2) -> U15#(isNat(V1),V2) 0.00/0.39 U14#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U15#(tt,V2) -> U16#(isNat(V2)) 0.00/0.39 U15#(tt,V2) -> ISNAT(V2) 0.00/0.39 U21#(tt,V1) -> U22#(isNatKind(V1),V1) 0.00/0.39 U21#(tt,V1) -> ISNATKIND(V1) 0.00/0.39 U22#(tt,V1) -> U23#(isNat(V1)) 0.00/0.39 U22#(tt,V1) -> ISNAT(V1) 0.00/0.39 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.39 U31#(tt,V1,V2) -> ISNATKIND(V1) 0.00/0.39 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.39 U32#(tt,V1,V2) -> ISNATKIND(V2) 0.00/0.39 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.39 U33#(tt,V1,V2) -> ISNATKIND(V2) 0.00/0.39 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.39 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U35#(tt,V2) -> U36#(isNat(V2)) 0.00/0.39 U35#(tt,V2) -> ISNAT(V2) 0.00/0.39 U41#(tt,V2) -> U42#(isNatKind(V2)) 0.00/0.39 U41#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 U61#(tt,V2) -> U62#(isNatKind(V2)) 0.00/0.39 U61#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 U71#(tt,N) -> U72#(isNatKind(N),N) 0.00/0.39 U71#(tt,N) -> ISNATKIND(N) 0.00/0.39 U72#(tt,N) -> N 0.00/0.39 U81#(tt,M,N) -> U82#(isNatKind(M),M,N) 0.00/0.39 U81#(tt,M,N) -> ISNATKIND(M) 0.00/0.39 U82#(tt,M,N) -> U83#(isNat(N),M,N) 0.00/0.39 U82#(tt,M,N) -> ISNAT(N) 0.00/0.39 U83#(tt,M,N) -> U84#(isNatKind(N),M,N) 0.00/0.39 U83#(tt,M,N) -> ISNATKIND(N) 0.00/0.39 U84#(tt,M,N) -> PLUS(N,M) 0.00/0.39 U84#(tt,M,N) -> M 0.00/0.39 U84#(tt,M,N) -> N 0.00/0.39 U91#(tt,N) -> U92#(isNatKind(N)) 0.00/0.39 U91#(tt,N) -> ISNATKIND(N) 0.00/0.39 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(plus(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(x(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1) 0.00/0.39 ISNAT(s(V1)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(plus(V1,V2)) -> U41#(isNatKind(V1),V2) 0.00/0.39 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(x(V1,V2)) -> U61#(isNatKind(V1),V2) 0.00/0.39 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(s(V1)) -> U51#(isNatKind(V1)) 0.00/0.39 ISNATKIND(s(V1)) -> ISNATKIND(V1) 0.00/0.39 PLUS(N,0) -> U71#(isNat(N),N) 0.00/0.39 PLUS(N,0) -> ISNAT(N) 0.00/0.39 PLUS(N,s(M)) -> U81#(isNat(M),M,N) 0.00/0.39 PLUS(N,s(M)) -> ISNAT(M) 0.00/0.39 X(N,0) -> U91#(isNat(N),N) 0.00/0.39 X(N,0) -> ISNAT(N) 0.00/0.39 X(N,s(M)) -> U101#(isNat(M),M,N) 0.00/0.39 X(N,s(M)) -> ISNAT(M) 0.00/0.39 -> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 -> Unhiding rules: 0.00/0.39 Empty 0.00/0.39 ->Strongly Connected Components: 0.00/0.39 ->->Cycle: 0.00/0.39 ->->-> Pairs: 0.00/0.39 U41#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 U61#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 ISNATKIND(plus(V1,V2)) -> U41#(isNatKind(V1),V2) 0.00/0.39 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(x(V1,V2)) -> U61#(isNatKind(V1),V2) 0.00/0.39 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(s(V1)) -> ISNATKIND(V1) 0.00/0.39 ->->-> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 ->->-> Unhiding rules: 0.00/0.39 Empty 0.00/0.39 ->->Cycle: 0.00/0.39 ->->-> Pairs: 0.00/0.39 U11#(tt,V1,V2) -> U12#(isNatKind(V1),V1,V2) 0.00/0.39 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2) 0.00/0.39 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2) 0.00/0.39 U14#(tt,V1,V2) -> U15#(isNat(V1),V2) 0.00/0.39 U14#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U15#(tt,V2) -> ISNAT(V2) 0.00/0.39 U21#(tt,V1) -> U22#(isNatKind(V1),V1) 0.00/0.39 U22#(tt,V1) -> ISNAT(V1) 0.00/0.39 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.39 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.39 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.39 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.39 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U35#(tt,V2) -> ISNAT(V2) 0.00/0.39 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1) 0.00/0.39 ->->-> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 ->->-> Unhiding rules: 0.00/0.39 Empty 0.00/0.39 ->->Cycle: 0.00/0.39 ->->-> Pairs: 0.00/0.39 U81#(tt,M,N) -> U82#(isNatKind(M),M,N) 0.00/0.39 U82#(tt,M,N) -> U83#(isNat(N),M,N) 0.00/0.39 U83#(tt,M,N) -> U84#(isNatKind(N),M,N) 0.00/0.39 U84#(tt,M,N) -> PLUS(N,M) 0.00/0.39 PLUS(N,s(M)) -> U81#(isNat(M),M,N) 0.00/0.39 ->->-> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 ->->-> Unhiding rules: 0.00/0.39 Empty 0.00/0.39 ->->Cycle: 0.00/0.39 ->->-> Pairs: 0.00/0.39 U101#(tt,M,N) -> U102#(isNatKind(M),M,N) 0.00/0.39 U102#(tt,M,N) -> U103#(isNat(N),M,N) 0.00/0.39 U103#(tt,M,N) -> U104#(isNatKind(N),M,N) 0.00/0.39 U104#(tt,M,N) -> X(N,M) 0.00/0.39 X(N,s(M)) -> U101#(isNat(M),M,N) 0.00/0.39 ->->-> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 ->->-> Unhiding rules: 0.00/0.39 Empty 0.00/0.39 0.00/0.39 0.00/0.39 The problem is decomposed in 4 subproblems. 0.00/0.39 0.00/0.39 Problem 1.1: 0.00/0.39 0.00/0.39 SubNColl Processor: 0.00/0.39 -> Pairs: 0.00/0.39 U41#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 U61#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 ISNATKIND(plus(V1,V2)) -> U41#(isNatKind(V1),V2) 0.00/0.39 ISNATKIND(plus(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(x(V1,V2)) -> U61#(isNatKind(V1),V2) 0.00/0.39 ISNATKIND(x(V1,V2)) -> ISNATKIND(V1) 0.00/0.39 ISNATKIND(s(V1)) -> ISNATKIND(V1) 0.00/0.39 -> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 -> Unhiding rules: 0.00/0.39 Empty 0.00/0.39 ->Projection: 0.00/0.39 pi(U41#) = 2 0.00/0.39 pi(U61#) = 2 0.00/0.39 pi(ISNATKIND) = 1 0.00/0.39 0.00/0.39 Problem 1.1: 0.00/0.39 0.00/0.39 SCC Processor: 0.00/0.39 -> Pairs: 0.00/0.39 U41#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 U61#(tt,V2) -> ISNATKIND(V2) 0.00/0.39 -> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 -> Unhiding rules: 0.00/0.39 Empty 0.00/0.39 ->Strongly Connected Components: 0.00/0.39 There is no strongly connected component 0.00/0.39 0.00/0.39 The problem is finite. 0.00/0.39 0.00/0.39 Problem 1.2: 0.00/0.39 0.00/0.39 Reduction Pairs Processor: 0.00/0.39 -> Pairs: 0.00/0.39 U11#(tt,V1,V2) -> U12#(isNatKind(V1),V1,V2) 0.00/0.39 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2) 0.00/0.39 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2) 0.00/0.39 U14#(tt,V1,V2) -> U15#(isNat(V1),V2) 0.00/0.39 U14#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U15#(tt,V2) -> ISNAT(V2) 0.00/0.39 U21#(tt,V1) -> U22#(isNatKind(V1),V1) 0.00/0.39 U22#(tt,V1) -> ISNAT(V1) 0.00/0.39 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.39 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.39 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.39 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.39 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U35#(tt,V2) -> ISNAT(V2) 0.00/0.39 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1) 0.00/0.39 -> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 plus(N,0) -> U71(isNat(N),N) 0.00/0.39 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.39 x(N,0) -> U91(isNat(N),N) 0.00/0.39 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.39 -> Unhiding rules: 0.00/0.39 Empty 0.00/0.39 -> Usable rules: 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.39 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.39 ->Interpretation type: 0.00/0.39 Linear 0.00/0.39 ->Coefficients: 0.00/0.39 Natural Numbers 0.00/0.39 ->Dimension: 0.00/0.39 1 0.00/0.39 ->Bound: 0.00/0.39 2 0.00/0.39 ->Interpretation: 0.00/0.39 0.00/0.39 [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 0.00/0.39 [U12](X1,X2,X3) = X1 + 2.X2 + X3 + 2 0.00/0.39 [U13](X1,X2,X3) = X1 + 2.X2 + X3 + 2 0.00/0.39 [U14](X1,X2,X3) = 2.X1 + 2.X2 + X3 0.00/0.39 [U15](X1,X2) = 2.X1 + X2 0.00/0.39 [U16](X) = X + 2 0.00/0.39 [U21](X1,X2) = X1 + 2.X2 + 2 0.00/0.39 [U22](X1,X2) = X1 + 2.X2 + 2 0.00/0.39 [U23](X) = X + 2 0.00/0.39 [U31](X1,X2,X3) = X2 + 2.X3 + 2 0.00/0.39 [U32](X1,X2,X3) = X1 + X2 + 2.X3 0.00/0.39 [U33](X1,X2,X3) = X1 + X2 + 2.X3 0.00/0.39 [U34](X1,X2,X3) = X2 + X3 + 2 0.00/0.39 [U35](X1,X2) = X2 + 2 0.00/0.39 [U36](X) = X 0.00/0.39 [U41](X1,X2) = X1 0.00/0.39 [U42](X) = 2 0.00/0.39 [U51](X) = 2 0.00/0.39 [U61](X1,X2) = 2 0.00/0.39 [U62](X) = X 0.00/0.39 [isNat](X) = X + 2 0.00/0.39 [isNatKind](X) = 2 0.00/0.39 [plus](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.39 [x](X1,X2) = 2.X1 + 2.X2 0.00/0.39 [0] = 0 0.00/0.39 [s](X) = 2.X + 2 0.00/0.39 [tt] = 2 0.00/0.39 [U11#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2 0.00/0.39 [U12#](X1,X2,X3) = 2.X2 + 2.X3 + 2 0.00/0.39 [U13#](X1,X2,X3) = X1 + 2.X2 + 2.X3 0.00/0.39 [U14#](X1,X2,X3) = 2.X2 + 2.X3 + 2 0.00/0.39 [U15#](X1,X2) = 2.X2 + 2 0.00/0.39 [U21#](X1,X2) = 2.X2 + 2 0.00/0.39 [U22#](X1,X2) = 2.X2 + 2 0.00/0.39 [U31#](X1,X2,X3) = X1 + 2.X2 + 2.X3 0.00/0.39 [U32#](X1,X2,X3) = X1 + 2.X2 + 2.X3 0.00/0.39 [U33#](X1,X2,X3) = 2.X2 + 2.X3 + 2 0.00/0.39 [U34#](X1,X2,X3) = 2.X2 + 2.X3 + 2 0.00/0.39 [U35#](X1,X2) = X1 + 2.X2 0.00/0.39 [ISNAT](X) = 2.X + 2 0.00/0.39 0.00/0.39 Problem 1.2: 0.00/0.39 0.00/0.39 SCC Processor: 0.00/0.39 -> Pairs: 0.00/0.39 U12#(tt,V1,V2) -> U13#(isNatKind(V2),V1,V2) 0.00/0.39 U13#(tt,V1,V2) -> U14#(isNatKind(V2),V1,V2) 0.00/0.39 U14#(tt,V1,V2) -> U15#(isNat(V1),V2) 0.00/0.39 U14#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U15#(tt,V2) -> ISNAT(V2) 0.00/0.39 U21#(tt,V1) -> U22#(isNatKind(V1),V1) 0.00/0.39 U22#(tt,V1) -> ISNAT(V1) 0.00/0.39 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.39 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.39 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.39 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.39 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.39 U35#(tt,V2) -> ISNAT(V2) 0.00/0.39 ISNAT(plus(V1,V2)) -> U11#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.39 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1) 0.00/0.39 -> Rules: 0.00/0.39 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.39 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.39 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.39 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.39 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.39 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.39 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.39 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.39 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.39 U16(tt) -> tt 0.00/0.39 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.39 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.39 U23(tt) -> tt 0.00/0.39 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.39 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.39 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.39 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.39 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.39 U36(tt) -> tt 0.00/0.39 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.39 U42(tt) -> tt 0.00/0.39 U51(tt) -> tt 0.00/0.39 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.39 U62(tt) -> tt 0.00/0.39 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.39 U72(tt,N) -> N 0.00/0.39 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.39 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.39 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.39 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.39 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.39 U92(tt) -> 0 0.00/0.39 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.39 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.39 isNat(0) -> tt 0.00/0.39 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.39 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.39 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.39 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 -> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 ->Strongly Connected Components: 0.00/0.40 ->->Cycle: 0.00/0.40 ->->-> Pairs: 0.00/0.40 U21#(tt,V1) -> U22#(isNatKind(V1),V1) 0.00/0.40 U22#(tt,V1) -> ISNAT(V1) 0.00/0.40 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.40 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.40 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.40 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.40 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.40 U35#(tt,V2) -> ISNAT(V2) 0.00/0.40 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.40 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1) 0.00/0.40 ->->-> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 ->->-> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 0.00/0.40 Problem 1.2: 0.00/0.40 0.00/0.40 Reduction Pairs Processor: 0.00/0.40 -> Pairs: 0.00/0.40 U21#(tt,V1) -> U22#(isNatKind(V1),V1) 0.00/0.40 U22#(tt,V1) -> ISNAT(V1) 0.00/0.40 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.40 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.40 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.40 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.40 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.40 U35#(tt,V2) -> ISNAT(V2) 0.00/0.40 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.40 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1) 0.00/0.40 -> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 -> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 -> Usable rules: 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 ->Interpretation type: 0.00/0.40 Linear 0.00/0.40 ->Coefficients: 0.00/0.40 Natural Numbers 0.00/0.40 ->Dimension: 0.00/0.40 1 0.00/0.40 ->Bound: 0.00/0.40 2 0.00/0.40 ->Interpretation: 0.00/0.40 0.00/0.40 [U11](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 0.00/0.40 [U12](X1,X2,X3) = X1 + 2.X3 + 1 0.00/0.40 [U13](X1,X2,X3) = 2.X3 + 2 0.00/0.40 [U14](X1,X2,X3) = X1 + 2 0.00/0.40 [U15](X1,X2) = 2 0.00/0.40 [U16](X) = 2 0.00/0.40 [U21](X1,X2) = X1 + 2.X2 + 2 0.00/0.40 [U22](X1,X2) = X1 + 2 0.00/0.40 [U23](X) = 2 0.00/0.40 [U31](X1,X2,X3) = 2.X1 + 2.X3 + 2 0.00/0.40 [U32](X1,X2,X3) = 2.X3 + 2 0.00/0.40 [U33](X1,X2,X3) = 2.X3 + 2 0.00/0.40 [U34](X1,X2,X3) = X1 + 2 0.00/0.40 [U35](X1,X2) = 2 0.00/0.40 [U36](X) = 2 0.00/0.40 [U41](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.40 [U42](X) = X 0.00/0.40 [U51](X) = 2.X 0.00/0.40 [U61](X1,X2) = 2.X1 + 2.X2 0.00/0.40 [U62](X) = X + 2 0.00/0.40 [isNat](X) = 2.X 0.00/0.40 [isNatKind](X) = 2.X 0.00/0.40 [plus](X1,X2) = 2.X1 + 2.X2 + 1 0.00/0.40 [x](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.40 [0] = 2 0.00/0.40 [s](X) = 2.X + 2 0.00/0.40 [tt] = 2 0.00/0.40 [U21#](X1,X2) = X1 + 2.X2 + 2 0.00/0.40 [U22#](X1,X2) = 2.X2 + 2 0.00/0.40 [U31#](X1,X2,X3) = 2.X2 + 2.X3 + 2 0.00/0.40 [U32#](X1,X2,X3) = 2.X2 + 2.X3 + 2 0.00/0.40 [U33#](X1,X2,X3) = 2.X2 + 2.X3 + 2 0.00/0.40 [U34#](X1,X2,X3) = 2.X2 + 2.X3 + 2 0.00/0.40 [U35#](X1,X2) = X1 + 2.X2 + 2 0.00/0.40 [ISNAT](X) = 2.X + 2 0.00/0.40 0.00/0.40 Problem 1.2: 0.00/0.40 0.00/0.40 SCC Processor: 0.00/0.40 -> Pairs: 0.00/0.40 U22#(tt,V1) -> ISNAT(V1) 0.00/0.40 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.40 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.40 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.40 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.40 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.40 U35#(tt,V2) -> ISNAT(V2) 0.00/0.40 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.40 ISNAT(s(V1)) -> U21#(isNatKind(V1),V1) 0.00/0.40 -> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 -> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 ->Strongly Connected Components: 0.00/0.40 ->->Cycle: 0.00/0.40 ->->-> Pairs: 0.00/0.40 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.40 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.40 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.40 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.40 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.40 U35#(tt,V2) -> ISNAT(V2) 0.00/0.40 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.40 ->->-> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 ->->-> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 0.00/0.40 Problem 1.2: 0.00/0.40 0.00/0.40 Reduction Pairs Processor: 0.00/0.40 -> Pairs: 0.00/0.40 U31#(tt,V1,V2) -> U32#(isNatKind(V1),V1,V2) 0.00/0.40 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.40 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.40 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.40 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.40 U35#(tt,V2) -> ISNAT(V2) 0.00/0.40 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.40 -> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 -> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 -> Usable rules: 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 ->Interpretation type: 0.00/0.40 Linear 0.00/0.40 ->Coefficients: 0.00/0.40 Natural Numbers 0.00/0.40 ->Dimension: 0.00/0.40 1 0.00/0.40 ->Bound: 0.00/0.40 2 0.00/0.40 ->Interpretation: 0.00/0.40 0.00/0.40 [U11](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 0.00/0.40 [U12](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 0.00/0.40 [U13](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 0.00/0.40 [U14](X1,X2,X3) = X1 + 2.X2 + 2 0.00/0.40 [U15](X1,X2) = X1 0.00/0.40 [U16](X) = 2 0.00/0.40 [U21](X1,X2) = X1 0.00/0.40 [U22](X1,X2) = 2 0.00/0.40 [U23](X) = 2 0.00/0.40 [U31](X1,X2,X3) = 2.X1 + 2.X3 + 2 0.00/0.40 [U32](X1,X2,X3) = 2.X1 + 2.X3 + 2 0.00/0.40 [U33](X1,X2,X3) = 2.X1 + 2.X3 + 2 0.00/0.40 [U34](X1,X2,X3) = X1 + 2.X3 + 2 0.00/0.40 [U35](X1,X2) = 2.X2 + 2 0.00/0.40 [U36](X) = X 0.00/0.40 [U41](X1,X2) = 2 0.00/0.40 [U42](X) = X 0.00/0.40 [U51](X) = X 0.00/0.40 [U61](X1,X2) = 2 0.00/0.40 [U62](X) = 2 0.00/0.40 [isNat](X) = 2.X + 2 0.00/0.40 [isNatKind](X) = 2 0.00/0.40 [plus](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.40 [x](X1,X2) = 2.X1 + 2.X2 + 2 0.00/0.40 [0] = 0 0.00/0.40 [s](X) = 2 0.00/0.40 [tt] = 2 0.00/0.40 [U31#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 1 0.00/0.40 [U32#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 0.00/0.40 [U33#](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 0.00/0.40 [U34#](X1,X2,X3) = X1 + 2.X2 + 2.X3 + 2 0.00/0.40 [U35#](X1,X2) = X1 + 2.X2 + 2 0.00/0.40 [ISNAT](X) = 2.X + 1 0.00/0.40 0.00/0.40 Problem 1.2: 0.00/0.40 0.00/0.40 SCC Processor: 0.00/0.40 -> Pairs: 0.00/0.40 U32#(tt,V1,V2) -> U33#(isNatKind(V2),V1,V2) 0.00/0.40 U33#(tt,V1,V2) -> U34#(isNatKind(V2),V1,V2) 0.00/0.40 U34#(tt,V1,V2) -> U35#(isNat(V1),V2) 0.00/0.40 U34#(tt,V1,V2) -> ISNAT(V1) 0.00/0.40 U35#(tt,V2) -> ISNAT(V2) 0.00/0.40 ISNAT(x(V1,V2)) -> U31#(isNatKind(V1),V1,V2) 0.00/0.40 -> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 -> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 ->Strongly Connected Components: 0.00/0.40 There is no strongly connected component 0.00/0.40 0.00/0.40 The problem is finite. 0.00/0.40 0.00/0.40 Problem 1.3: 0.00/0.40 0.00/0.40 SubNColl Processor: 0.00/0.40 -> Pairs: 0.00/0.40 U81#(tt,M,N) -> U82#(isNatKind(M),M,N) 0.00/0.40 U82#(tt,M,N) -> U83#(isNat(N),M,N) 0.00/0.40 U83#(tt,M,N) -> U84#(isNatKind(N),M,N) 0.00/0.40 U84#(tt,M,N) -> PLUS(N,M) 0.00/0.40 PLUS(N,s(M)) -> U81#(isNat(M),M,N) 0.00/0.40 -> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 -> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 ->Projection: 0.00/0.40 pi(U81#) = 2 0.00/0.40 pi(U82#) = 2 0.00/0.40 pi(U83#) = 2 0.00/0.40 pi(U84#) = 2 0.00/0.40 pi(PLUS) = 2 0.00/0.40 0.00/0.40 Problem 1.3: 0.00/0.40 0.00/0.40 SCC Processor: 0.00/0.40 -> Pairs: 0.00/0.40 U81#(tt,M,N) -> U82#(isNatKind(M),M,N) 0.00/0.40 U82#(tt,M,N) -> U83#(isNat(N),M,N) 0.00/0.40 U83#(tt,M,N) -> U84#(isNatKind(N),M,N) 0.00/0.40 U84#(tt,M,N) -> PLUS(N,M) 0.00/0.40 -> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 -> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 ->Strongly Connected Components: 0.00/0.40 There is no strongly connected component 0.00/0.40 0.00/0.40 The problem is finite. 0.00/0.40 0.00/0.40 Problem 1.4: 0.00/0.40 0.00/0.40 SubNColl Processor: 0.00/0.40 -> Pairs: 0.00/0.40 U101#(tt,M,N) -> U102#(isNatKind(M),M,N) 0.00/0.40 U102#(tt,M,N) -> U103#(isNat(N),M,N) 0.00/0.40 U103#(tt,M,N) -> U104#(isNatKind(N),M,N) 0.00/0.40 U104#(tt,M,N) -> X(N,M) 0.00/0.40 X(N,s(M)) -> U101#(isNat(M),M,N) 0.00/0.40 -> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 -> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 ->Projection: 0.00/0.40 pi(U101#) = 2 0.00/0.40 pi(U102#) = 2 0.00/0.40 pi(U103#) = 2 0.00/0.40 pi(U104#) = 2 0.00/0.40 pi(X) = 2 0.00/0.40 0.00/0.40 Problem 1.4: 0.00/0.40 0.00/0.40 SCC Processor: 0.00/0.40 -> Pairs: 0.00/0.40 U101#(tt,M,N) -> U102#(isNatKind(M),M,N) 0.00/0.40 U102#(tt,M,N) -> U103#(isNat(N),M,N) 0.00/0.40 U103#(tt,M,N) -> U104#(isNatKind(N),M,N) 0.00/0.40 U104#(tt,M,N) -> X(N,M) 0.00/0.40 -> Rules: 0.00/0.40 U101(tt,M,N) -> U102(isNatKind(M),M,N) 0.00/0.40 U102(tt,M,N) -> U103(isNat(N),M,N) 0.00/0.40 U103(tt,M,N) -> U104(isNatKind(N),M,N) 0.00/0.40 U104(tt,M,N) -> plus(x(N,M),N) 0.00/0.40 U11(tt,V1,V2) -> U12(isNatKind(V1),V1,V2) 0.00/0.40 U12(tt,V1,V2) -> U13(isNatKind(V2),V1,V2) 0.00/0.40 U13(tt,V1,V2) -> U14(isNatKind(V2),V1,V2) 0.00/0.40 U14(tt,V1,V2) -> U15(isNat(V1),V2) 0.00/0.40 U15(tt,V2) -> U16(isNat(V2)) 0.00/0.40 U16(tt) -> tt 0.00/0.40 U21(tt,V1) -> U22(isNatKind(V1),V1) 0.00/0.40 U22(tt,V1) -> U23(isNat(V1)) 0.00/0.40 U23(tt) -> tt 0.00/0.40 U31(tt,V1,V2) -> U32(isNatKind(V1),V1,V2) 0.00/0.40 U32(tt,V1,V2) -> U33(isNatKind(V2),V1,V2) 0.00/0.40 U33(tt,V1,V2) -> U34(isNatKind(V2),V1,V2) 0.00/0.40 U34(tt,V1,V2) -> U35(isNat(V1),V2) 0.00/0.40 U35(tt,V2) -> U36(isNat(V2)) 0.00/0.40 U36(tt) -> tt 0.00/0.40 U41(tt,V2) -> U42(isNatKind(V2)) 0.00/0.40 U42(tt) -> tt 0.00/0.40 U51(tt) -> tt 0.00/0.40 U61(tt,V2) -> U62(isNatKind(V2)) 0.00/0.40 U62(tt) -> tt 0.00/0.40 U71(tt,N) -> U72(isNatKind(N),N) 0.00/0.40 U72(tt,N) -> N 0.00/0.40 U81(tt,M,N) -> U82(isNatKind(M),M,N) 0.00/0.40 U82(tt,M,N) -> U83(isNat(N),M,N) 0.00/0.40 U83(tt,M,N) -> U84(isNatKind(N),M,N) 0.00/0.40 U84(tt,M,N) -> s(plus(N,M)) 0.00/0.40 U91(tt,N) -> U92(isNatKind(N)) 0.00/0.40 U92(tt) -> 0 0.00/0.40 isNat(plus(V1,V2)) -> U11(isNatKind(V1),V1,V2) 0.00/0.40 isNat(x(V1,V2)) -> U31(isNatKind(V1),V1,V2) 0.00/0.40 isNat(0) -> tt 0.00/0.40 isNat(s(V1)) -> U21(isNatKind(V1),V1) 0.00/0.40 isNatKind(plus(V1,V2)) -> U41(isNatKind(V1),V2) 0.00/0.40 isNatKind(x(V1,V2)) -> U61(isNatKind(V1),V2) 0.00/0.40 isNatKind(0) -> tt 0.00/0.40 isNatKind(s(V1)) -> U51(isNatKind(V1)) 0.00/0.40 plus(N,0) -> U71(isNat(N),N) 0.00/0.40 plus(N,s(M)) -> U81(isNat(M),M,N) 0.00/0.40 x(N,0) -> U91(isNat(N),N) 0.00/0.40 x(N,s(M)) -> U101(isNat(M),M,N) 0.00/0.40 -> Unhiding rules: 0.00/0.40 Empty 0.00/0.40 ->Strongly Connected Components: 0.00/0.40 There is no strongly connected component 0.00/0.40 0.00/0.40 The problem is finite. 0.00/0.40 EOF