MAYBE 0.28/0.30 MAYBE 0.28/0.30 0.28/0.30 Problem: 0.28/0.30 active(U11(tt(),V1,V2)) -> mark(U12(isNatKind(V1),V1,V2)) 0.28/0.30 active(U12(tt(),V1,V2)) -> mark(U13(isNatKind(V2),V1,V2)) 0.28/0.30 active(U13(tt(),V1,V2)) -> mark(U14(isNatKind(V2),V1,V2)) 0.28/0.30 active(U14(tt(),V1,V2)) -> mark(U15(isNat(V1),V2)) 0.28/0.30 active(U15(tt(),V2)) -> mark(U16(isNat(V2))) 0.28/0.30 active(U16(tt())) -> mark(tt()) 0.28/0.30 active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1)) 0.28/0.30 active(U22(tt(),V1)) -> mark(U23(isNat(V1))) 0.28/0.30 active(U23(tt())) -> mark(tt()) 0.28/0.30 active(U31(tt(),V2)) -> mark(U32(isNatKind(V2))) 0.28/0.30 active(U32(tt())) -> mark(tt()) 0.28/0.30 active(U41(tt())) -> mark(tt()) 0.28/0.30 active(U51(tt(),N)) -> mark(U52(isNatKind(N),N)) 0.28/0.30 active(U52(tt(),N)) -> mark(N) 0.28/0.30 active(U61(tt(),M,N)) -> mark(U62(isNatKind(M),M,N)) 0.28/0.30 active(U62(tt(),M,N)) -> mark(U63(isNat(N),M,N)) 0.28/0.30 active(U63(tt(),M,N)) -> mark(U64(isNatKind(N),M,N)) 0.28/0.30 active(U64(tt(),M,N)) -> mark(s(plus(N,M))) 0.28/0.30 active(isNat(0())) -> mark(tt()) 0.28/0.30 active(isNat(plus(V1,V2))) -> mark(U11(isNatKind(V1),V1,V2)) 0.28/0.30 active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1)) 0.28/0.30 active(isNatKind(0())) -> mark(tt()) 0.28/0.30 active(isNatKind(plus(V1,V2))) -> mark(U31(isNatKind(V1),V2)) 0.28/0.30 active(isNatKind(s(V1))) -> mark(U41(isNatKind(V1))) 0.28/0.30 active(plus(N,0())) -> mark(U51(isNat(N),N)) 0.28/0.30 active(plus(N,s(M))) -> mark(U61(isNat(M),M,N)) 0.28/0.30 active(U11(X1,X2,X3)) -> U11(active(X1),X2,X3) 0.28/0.30 active(U12(X1,X2,X3)) -> U12(active(X1),X2,X3) 0.28/0.30 active(U13(X1,X2,X3)) -> U13(active(X1),X2,X3) 0.28/0.30 active(U14(X1,X2,X3)) -> U14(active(X1),X2,X3) 0.28/0.30 active(U15(X1,X2)) -> U15(active(X1),X2) 0.28/0.30 active(U16(X)) -> U16(active(X)) 0.28/0.30 active(U21(X1,X2)) -> U21(active(X1),X2) 0.28/0.30 active(U22(X1,X2)) -> U22(active(X1),X2) 0.28/0.30 active(U23(X)) -> U23(active(X)) 0.28/0.30 active(U31(X1,X2)) -> U31(active(X1),X2) 0.28/0.30 active(U32(X)) -> U32(active(X)) 0.28/0.30 active(U41(X)) -> U41(active(X)) 0.28/0.30 active(U51(X1,X2)) -> U51(active(X1),X2) 0.28/0.30 active(U52(X1,X2)) -> U52(active(X1),X2) 0.28/0.30 active(U61(X1,X2,X3)) -> U61(active(X1),X2,X3) 0.28/0.30 active(U62(X1,X2,X3)) -> U62(active(X1),X2,X3) 0.28/0.30 active(U63(X1,X2,X3)) -> U63(active(X1),X2,X3) 0.28/0.30 active(U64(X1,X2,X3)) -> U64(active(X1),X2,X3) 0.28/0.30 active(s(X)) -> s(active(X)) 0.28/0.30 active(plus(X1,X2)) -> plus(active(X1),X2) 0.28/0.30 active(plus(X1,X2)) -> plus(X1,active(X2)) 0.28/0.30 U11(mark(X1),X2,X3) -> mark(U11(X1,X2,X3)) 0.28/0.30 U12(mark(X1),X2,X3) -> mark(U12(X1,X2,X3)) 0.28/0.30 U13(mark(X1),X2,X3) -> mark(U13(X1,X2,X3)) 0.28/0.30 U14(mark(X1),X2,X3) -> mark(U14(X1,X2,X3)) 0.28/0.30 U15(mark(X1),X2) -> mark(U15(X1,X2)) 0.28/0.30 U16(mark(X)) -> mark(U16(X)) 0.28/0.30 U21(mark(X1),X2) -> mark(U21(X1,X2)) 0.28/0.30 U22(mark(X1),X2) -> mark(U22(X1,X2)) 0.28/0.30 U23(mark(X)) -> mark(U23(X)) 0.28/0.30 U31(mark(X1),X2) -> mark(U31(X1,X2)) 0.28/0.30 U32(mark(X)) -> mark(U32(X)) 0.28/0.30 U41(mark(X)) -> mark(U41(X)) 0.28/0.30 U51(mark(X1),X2) -> mark(U51(X1,X2)) 0.28/0.30 U52(mark(X1),X2) -> mark(U52(X1,X2)) 0.28/0.30 U61(mark(X1),X2,X3) -> mark(U61(X1,X2,X3)) 0.28/0.30 U62(mark(X1),X2,X3) -> mark(U62(X1,X2,X3)) 0.28/0.30 U63(mark(X1),X2,X3) -> mark(U63(X1,X2,X3)) 0.28/0.30 U64(mark(X1),X2,X3) -> mark(U64(X1,X2,X3)) 0.28/0.30 s(mark(X)) -> mark(s(X)) 0.28/0.30 plus(mark(X1),X2) -> mark(plus(X1,X2)) 0.28/0.30 plus(X1,mark(X2)) -> mark(plus(X1,X2)) 0.28/0.30 proper(U11(X1,X2,X3)) -> U11(proper(X1),proper(X2),proper(X3)) 0.28/0.30 proper(tt()) -> ok(tt()) 0.28/0.30 proper(U12(X1,X2,X3)) -> U12(proper(X1),proper(X2),proper(X3)) 0.28/0.30 proper(isNatKind(X)) -> isNatKind(proper(X)) 0.28/0.30 proper(U13(X1,X2,X3)) -> U13(proper(X1),proper(X2),proper(X3)) 0.28/0.30 proper(U14(X1,X2,X3)) -> U14(proper(X1),proper(X2),proper(X3)) 0.28/0.30 proper(U15(X1,X2)) -> U15(proper(X1),proper(X2)) 0.28/0.30 proper(isNat(X)) -> isNat(proper(X)) 0.28/0.30 proper(U16(X)) -> U16(proper(X)) 0.28/0.30 proper(U21(X1,X2)) -> U21(proper(X1),proper(X2)) 0.28/0.30 proper(U22(X1,X2)) -> U22(proper(X1),proper(X2)) 0.28/0.30 proper(U23(X)) -> U23(proper(X)) 0.28/0.30 proper(U31(X1,X2)) -> U31(proper(X1),proper(X2)) 0.28/0.30 proper(U32(X)) -> U32(proper(X)) 0.28/0.30 proper(U41(X)) -> U41(proper(X)) 0.28/0.30 proper(U51(X1,X2)) -> U51(proper(X1),proper(X2)) 0.28/0.30 proper(U52(X1,X2)) -> U52(proper(X1),proper(X2)) 0.28/0.30 proper(U61(X1,X2,X3)) -> U61(proper(X1),proper(X2),proper(X3)) 0.28/0.30 proper(U62(X1,X2,X3)) -> U62(proper(X1),proper(X2),proper(X3)) 0.28/0.30 proper(U63(X1,X2,X3)) -> U63(proper(X1),proper(X2),proper(X3)) 0.28/0.30 proper(U64(X1,X2,X3)) -> U64(proper(X1),proper(X2),proper(X3)) 0.28/0.30 proper(s(X)) -> s(proper(X)) 0.28/0.30 proper(plus(X1,X2)) -> plus(proper(X1),proper(X2)) 0.28/0.30 proper(0()) -> ok(0()) 0.28/0.30 U11(ok(X1),ok(X2),ok(X3)) -> ok(U11(X1,X2,X3)) 0.28/0.30 U12(ok(X1),ok(X2),ok(X3)) -> ok(U12(X1,X2,X3)) 0.28/0.30 isNatKind(ok(X)) -> ok(isNatKind(X)) 0.28/0.30 U13(ok(X1),ok(X2),ok(X3)) -> ok(U13(X1,X2,X3)) 0.28/0.30 U14(ok(X1),ok(X2),ok(X3)) -> ok(U14(X1,X2,X3)) 0.28/0.30 U15(ok(X1),ok(X2)) -> ok(U15(X1,X2)) 0.28/0.30 isNat(ok(X)) -> ok(isNat(X)) 0.28/0.30 U16(ok(X)) -> ok(U16(X)) 0.28/0.30 U21(ok(X1),ok(X2)) -> ok(U21(X1,X2)) 0.28/0.30 U22(ok(X1),ok(X2)) -> ok(U22(X1,X2)) 0.28/0.30 U23(ok(X)) -> ok(U23(X)) 0.28/0.30 U31(ok(X1),ok(X2)) -> ok(U31(X1,X2)) 0.28/0.30 U32(ok(X)) -> ok(U32(X)) 0.28/0.30 U41(ok(X)) -> ok(U41(X)) 0.28/0.30 U51(ok(X1),ok(X2)) -> ok(U51(X1,X2)) 0.28/0.30 U52(ok(X1),ok(X2)) -> ok(U52(X1,X2)) 0.28/0.30 U61(ok(X1),ok(X2),ok(X3)) -> ok(U61(X1,X2,X3)) 0.28/0.30 U62(ok(X1),ok(X2),ok(X3)) -> ok(U62(X1,X2,X3)) 0.28/0.30 U63(ok(X1),ok(X2),ok(X3)) -> ok(U63(X1,X2,X3)) 0.28/0.30 U64(ok(X1),ok(X2),ok(X3)) -> ok(U64(X1,X2,X3)) 0.28/0.30 s(ok(X)) -> ok(s(X)) 0.28/0.30 plus(ok(X1),ok(X2)) -> ok(plus(X1,X2)) 0.28/0.30 top(mark(X)) -> top(proper(X)) 0.28/0.30 top(ok(X)) -> top(active(X)) 0.28/0.30 0.28/0.30 Proof: 0.28/0.30 Open 0.28/0.31 EOF