The TRS could not be proven terminating. The proof attempt took 60018 ms.
Problem 1 remains open; application of the following processors failed [DependencyGraph (timeout)].
| proper#(U92(X)) | → | U92#(proper(X)) | proper#(U41(X1, X2)) | → | proper#(X2) | |
| U34#(ok(X1), ok(X2), ok(X3)) | → | U34#(X1, X2, X3) | proper#(U41(X1, X2)) | → | proper#(X1) | |
| proper#(U11(X1, X2, X3)) | → | proper#(X3) | proper#(U15(X1, X2)) | → | proper#(X1) | |
| active#(U35(tt, V2)) | → | isNat#(V2) | proper#(U91(X1, X2)) | → | proper#(X2) | |
| active#(isNatKind(s(V1))) | → | U51#(isNatKind(V1)) | U51#(ok(X)) | → | U51#(X) | |
| active#(U15(tt, V2)) | → | U16#(isNat(V2)) | proper#(U31(X1, X2, X3)) | → | proper#(X2) | |
| proper#(U62(X)) | → | proper#(X) | U72#(mark(X1), X2) | → | U72#(X1, X2) | |
| proper#(U11(X1, X2, X3)) | → | proper#(X2) | proper#(U32(X1, X2, X3)) | → | proper#(X2) | |
| active#(U34(X1, X2, X3)) | → | U34#(active(X1), X2, X3) | active#(U34(tt, V1, V2)) | → | U35#(isNat(V1), V2) | |
| top#(mark(X)) | → | proper#(X) | active#(U13(X1, X2, X3)) | → | active#(X1) | |
| proper#(U72(X1, X2)) | → | proper#(X2) | isNatKind#(ok(X)) | → | isNatKind#(X) | |
| proper#(U104(X1, X2, X3)) | → | proper#(X2) | active#(U101(X1, X2, X3)) | → | U101#(active(X1), X2, X3) | |
| proper#(U71(X1, X2)) | → | U71#(proper(X1), proper(X2)) | active#(U81(X1, X2, X3)) | → | active#(X1) | |
| active#(U61(X1, X2)) | → | active#(X1) | proper#(U104(X1, X2, X3)) | → | proper#(X1) | |
| active#(U61(tt, V2)) | → | isNatKind#(V2) | U36#(mark(X)) | → | U36#(X) | |
| active#(U21(tt, V1)) | → | isNatKind#(V1) | active#(U12(X1, X2, X3)) | → | active#(X1) | |
| active#(U41(X1, X2)) | → | active#(X1) | proper#(U35(X1, X2)) | → | proper#(X2) | |
| proper#(U33(X1, X2, X3)) | → | U33#(proper(X1), proper(X2), proper(X3)) | proper#(U101(X1, X2, X3)) | → | proper#(X2) | |
| active#(plus(X1, X2)) | → | active#(X2) | active#(U72(X1, X2)) | → | active#(X1) | |
| proper#(U23(X)) | → | U23#(proper(X)) | proper#(U82(X1, X2, X3)) | → | proper#(X1) | |
| proper#(U12(X1, X2, X3)) | → | proper#(X1) | U102#(mark(X1), X2, X3) | → | U102#(X1, X2, X3) | |
| proper#(U104(X1, X2, X3)) | → | U104#(proper(X1), proper(X2), proper(X3)) | proper#(isNatKind(X)) | → | isNatKind#(proper(X)) | |
| active#(U15(tt, V2)) | → | isNat#(V2) | active#(U102(X1, X2, X3)) | → | U102#(active(X1), X2, X3) | |
| active#(U91(tt, N)) | → | isNatKind#(N) | U104#(ok(X1), ok(X2), ok(X3)) | → | U104#(X1, X2, X3) | |
| U15#(ok(X1), ok(X2)) | → | U15#(X1, X2) | active#(U34(X1, X2, X3)) | → | active#(X1) | |
| proper#(U16(X)) | → | U16#(proper(X)) | U14#(mark(X1), X2, X3) | → | U14#(X1, X2, X3) | |
| active#(U12(X1, X2, X3)) | → | U12#(active(X1), X2, X3) | active#(isNat(x(V1, V2))) | → | isNatKind#(V1) | |
| active#(U12(tt, V1, V2)) | → | U13#(isNatKind(V2), V1, V2) | active#(U22(X1, X2)) | → | active#(X1) | |
| active#(U83(X1, X2, X3)) | → | U83#(active(X1), X2, X3) | U101#(ok(X1), ok(X2), ok(X3)) | → | U101#(X1, X2, X3) | |
| proper#(U22(X1, X2)) | → | U22#(proper(X1), proper(X2)) | active#(U71(tt, N)) | → | U72#(isNatKind(N), N) | |
| plus#(ok(X1), ok(X2)) | → | plus#(X1, X2) | active#(U21(X1, X2)) | → | U21#(active(X1), X2) | |
| active#(U32(X1, X2, X3)) | → | U32#(active(X1), X2, X3) | proper#(U33(X1, X2, X3)) | → | proper#(X1) | |
| proper#(isNat(X)) | → | proper#(X) | proper#(U101(X1, X2, X3)) | → | proper#(X1) | |
| U15#(mark(X1), X2) | → | U15#(X1, X2) | U91#(ok(X1), ok(X2)) | → | U91#(X1, X2) | |
| active#(U83(X1, X2, X3)) | → | active#(X1) | U22#(ok(X1), ok(X2)) | → | U22#(X1, X2) | |
| proper#(U34(X1, X2, X3)) | → | proper#(X1) | active#(x(N, s(M))) | → | isNat#(M) | |
| active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | proper#(U13(X1, X2, X3)) | → | proper#(X1) | |
| proper#(U21(X1, X2)) | → | proper#(X1) | proper#(U103(X1, X2, X3)) | → | U103#(proper(X1), proper(X2), proper(X3)) | |
| active#(x(N, s(M))) | → | U101#(isNat(M), M, N) | active#(U104(tt, M, N)) | → | plus#(x(N, M), N) | |
| proper#(U13(X1, X2, X3)) | → | U13#(proper(X1), proper(X2), proper(X3)) | proper#(U32(X1, X2, X3)) | → | U32#(proper(X1), proper(X2), proper(X3)) | |
| proper#(U81(X1, X2, X3)) | → | U81#(proper(X1), proper(X2), proper(X3)) | active#(U22(X1, X2)) | → | U22#(active(X1), X2) | |
| proper#(U32(X1, X2, X3)) | → | proper#(X3) | proper#(U92(X)) | → | proper#(X) | |
| active#(U102(tt, M, N)) | → | U103#(isNat(N), M, N) | active#(U33(X1, X2, X3)) | → | active#(X1) | |
| proper#(U72(X1, X2)) | → | U72#(proper(X1), proper(X2)) | U71#(ok(X1), ok(X2)) | → | U71#(X1, X2) | |
| active#(isNat(plus(V1, V2))) | → | U11#(isNatKind(V1), V1, V2) | proper#(U71(X1, X2)) | → | proper#(X2) | |
| active#(U21(tt, V1)) | → | U22#(isNatKind(V1), V1) | active#(U33(X1, X2, X3)) | → | U33#(active(X1), X2, X3) | |
| active#(U41(X1, X2)) | → | U41#(active(X1), X2) | active#(U83(tt, M, N)) | → | isNatKind#(N) | |
| U16#(mark(X)) | → | U16#(X) | U34#(mark(X1), X2, X3) | → | U34#(X1, X2, X3) | |
| U91#(mark(X1), X2) | → | U91#(X1, X2) | proper#(U61(X1, X2)) | → | proper#(X2) | |
| U61#(mark(X1), X2) | → | U61#(X1, X2) | proper#(U14(X1, X2, X3)) | → | proper#(X1) | |
| proper#(plus(X1, X2)) | → | proper#(X1) | active#(U103(tt, M, N)) | → | U104#(isNatKind(N), M, N) | |
| active#(U101(X1, X2, X3)) | → | active#(X1) | proper#(U81(X1, X2, X3)) | → | proper#(X2) | |
| proper#(U104(X1, X2, X3)) | → | proper#(X3) | active#(U36(X)) | → | U36#(active(X)) | |
| proper#(U103(X1, X2, X3)) | → | proper#(X3) | U61#(ok(X1), ok(X2)) | → | U61#(X1, X2) | |
| proper#(U16(X)) | → | proper#(X) | U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | |
| proper#(U35(X1, X2)) | → | proper#(X1) | proper#(U12(X1, X2, X3)) | → | proper#(X3) | |
| active#(x(X1, X2)) | → | x#(X1, active(X2)) | active#(U22(tt, V1)) | → | U23#(isNat(V1)) | |
| proper#(s(X)) | → | proper#(X) | active#(plus(X1, X2)) | → | active#(X1) | |
| proper#(U33(X1, X2, X3)) | → | proper#(X3) | active#(U14(X1, X2, X3)) | → | active#(X1) | |
| U84#(ok(X1), ok(X2), ok(X3)) | → | U84#(X1, X2, X3) | active#(U13(X1, X2, X3)) | → | U13#(active(X1), X2, X3) | |
| proper#(U81(X1, X2, X3)) | → | proper#(X3) | active#(U82(X1, X2, X3)) | → | active#(X1) | |
| active#(U11(X1, X2, X3)) | → | U11#(active(X1), X2, X3) | active#(U103(X1, X2, X3)) | → | U103#(active(X1), X2, X3) | |
| U31#(ok(X1), ok(X2), ok(X3)) | → | U31#(X1, X2, X3) | active#(U16(X)) | → | U16#(active(X)) | |
| active#(x(X1, X2)) | → | x#(active(X1), X2) | proper#(U42(X)) | → | proper#(X) | |
| active#(U12(tt, V1, V2)) | → | isNatKind#(V2) | proper#(U15(X1, X2)) | → | proper#(X2) | |
| active#(isNat(plus(V1, V2))) | → | isNatKind#(V1) | proper#(U51(X)) | → | proper#(X) | |
| active#(U84(tt, M, N)) | → | plus#(N, M) | U41#(ok(X1), ok(X2)) | → | U41#(X1, X2) | |
| active#(U33(tt, V1, V2)) | → | U34#(isNatKind(V2), V1, V2) | proper#(U81(X1, X2, X3)) | → | proper#(X1) | |
| U41#(mark(X1), X2) | → | U41#(X1, X2) | proper#(plus(X1, X2)) | → | proper#(X2) | |
| active#(U104(tt, M, N)) | → | x#(N, M) | U103#(ok(X1), ok(X2), ok(X3)) | → | U103#(X1, X2, X3) | |
| proper#(U91(X1, X2)) | → | U91#(proper(X1), proper(X2)) | active#(U82(tt, M, N)) | → | U83#(isNat(N), M, N) | |
| active#(U71(X1, X2)) | → | active#(X1) | active#(U103(X1, X2, X3)) | → | active#(X1) | |
| proper#(U84(X1, X2, X3)) | → | proper#(X3) | proper#(U84(X1, X2, X3)) | → | proper#(X1) | |
| active#(isNat(x(V1, V2))) | → | U31#(isNatKind(V1), V1, V2) | U104#(mark(X1), X2, X3) | → | U104#(X1, X2, X3) | |
| proper#(U41(X1, X2)) | → | U41#(proper(X1), proper(X2)) | proper#(x(X1, X2)) | → | proper#(X2) | |
| active#(U102(tt, M, N)) | → | isNat#(N) | proper#(U11(X1, X2, X3)) | → | U11#(proper(X1), proper(X2), proper(X3)) | |
| U72#(ok(X1), ok(X2)) | → | U72#(X1, X2) | active#(U91(X1, X2)) | → | U91#(active(X1), X2) | |
| s#(mark(X)) | → | s#(X) | proper#(U84(X1, X2, X3)) | → | proper#(X2) | |
| active#(s(X)) | → | active#(X) | U14#(ok(X1), ok(X2), ok(X3)) | → | U14#(X1, X2, X3) | |
| U83#(mark(X1), X2, X3) | → | U83#(X1, X2, X3) | U32#(mark(X1), X2, X3) | → | U32#(X1, X2, X3) | |
| active#(U104(X1, X2, X3)) | → | U104#(active(X1), X2, X3) | active#(U102(X1, X2, X3)) | → | active#(X1) | |
| U35#(ok(X1), ok(X2)) | → | U35#(X1, X2) | U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) | |
| active#(U82(X1, X2, X3)) | → | U82#(active(X1), X2, X3) | active#(U101(tt, M, N)) | → | isNatKind#(M) | |
| active#(U35(tt, V2)) | → | U36#(isNat(V2)) | active#(isNatKind(plus(V1, V2))) | → | U41#(isNatKind(V1), V2) | |
| active#(U35(X1, X2)) | → | active#(X1) | active#(isNatKind(x(V1, V2))) | → | U61#(isNatKind(V1), V2) | |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V1) | active#(U15(X1, X2)) | → | active#(X1) | |
| active#(U104(X1, X2, X3)) | → | active#(X1) | active#(U84(tt, M, N)) | → | s#(plus(N, M)) | |
| active#(U13(tt, V1, V2)) | → | U14#(isNatKind(V2), V1, V2) | proper#(isNatKind(X)) | → | proper#(X) | |
| proper#(U42(X)) | → | U42#(proper(X)) | proper#(U33(X1, X2, X3)) | → | proper#(X2) | |
| active#(x(N, 0)) | → | U91#(isNat(N), N) | active#(U42(X)) | → | U42#(active(X)) | |
| x#(ok(X1), ok(X2)) | → | x#(X1, X2) | x#(X1, mark(X2)) | → | x#(X1, X2) | |
| proper#(U34(X1, X2, X3)) | → | U34#(proper(X1), proper(X2), proper(X3)) | active#(U32(X1, X2, X3)) | → | active#(X1) | |
| proper#(U23(X)) | → | proper#(X) | active#(plus(N, 0)) | → | isNat#(N) | |
| proper#(U13(X1, X2, X3)) | → | proper#(X2) | active#(U22(tt, V1)) | → | isNat#(V1) | |
| active#(U32(tt, V1, V2)) | → | isNatKind#(V2) | active#(U51(X)) | → | U51#(active(X)) | |
| U12#(mark(X1), X2, X3) | → | U12#(X1, X2, X3) | active#(U62(X)) | → | active#(X) | |
| isNat#(ok(X)) | → | isNat#(X) | active#(plus(N, s(M))) | → | U81#(isNat(M), M, N) | |
| active#(U35(X1, X2)) | → | U35#(active(X1), X2) | active#(isNat(s(V1))) | → | isNatKind#(V1) | |
| active#(U11(tt, V1, V2)) | → | isNatKind#(V1) | active#(U92(X)) | → | active#(X) | |
| active#(U103(tt, M, N)) | → | isNatKind#(N) | active#(U14(tt, V1, V2)) | → | U15#(isNat(V1), V2) | |
| active#(U31(X1, X2, X3)) | → | active#(X1) | proper#(U34(X1, X2, X3)) | → | proper#(X2) | |
| U11#(ok(X1), ok(X2), ok(X3)) | → | U11#(X1, X2, X3) | U33#(mark(X1), X2, X3) | → | U33#(X1, X2, X3) | |
| proper#(U101(X1, X2, X3)) | → | proper#(X3) | proper#(U62(X)) | → | U62#(proper(X)) | |
| U51#(mark(X)) | → | U51#(X) | plus#(mark(X1), X2) | → | plus#(X1, X2) | |
| U13#(mark(X1), X2, X3) | → | U13#(X1, X2, X3) | proper#(U15(X1, X2)) | → | U15#(proper(X1), proper(X2)) | |
| U103#(mark(X1), X2, X3) | → | U103#(X1, X2, X3) | active#(U15(X1, X2)) | → | U15#(active(X1), X2) | |
| active#(x(N, 0)) | → | isNat#(N) | active#(U31(tt, V1, V2)) | → | isNatKind#(V1) | |
| U22#(mark(X1), X2) | → | U22#(X1, X2) | U13#(ok(X1), ok(X2), ok(X3)) | → | U13#(X1, X2, X3) | |
| active#(U92(X)) | → | U92#(active(X)) | proper#(U103(X1, X2, X3)) | → | proper#(X2) | |
| top#(ok(X)) | → | active#(X) | active#(U91(X1, X2)) | → | active#(X1) | |
| U82#(mark(X1), X2, X3) | → | U82#(X1, X2, X3) | proper#(U103(X1, X2, X3)) | → | proper#(X1) | |
| x#(mark(X1), X2) | → | x#(X1, X2) | proper#(U91(X1, X2)) | → | proper#(X1) | |
| active#(U31(X1, X2, X3)) | → | U31#(active(X1), X2, X3) | U62#(mark(X)) | → | U62#(X) | |
| U83#(ok(X1), ok(X2), ok(X3)) | → | U83#(X1, X2, X3) | active#(U41(tt, V2)) | → | U42#(isNatKind(V2)) | |
| U71#(mark(X1), X2) | → | U71#(X1, X2) | proper#(U13(X1, X2, X3)) | → | proper#(X3) | |
| active#(U101(tt, M, N)) | → | U102#(isNatKind(M), M, N) | proper#(isNat(X)) | → | isNat#(proper(X)) | |
| active#(U33(tt, V1, V2)) | → | isNatKind#(V2) | U42#(ok(X)) | → | U42#(X) | |
| proper#(U102(X1, X2, X3)) | → | U102#(proper(X1), proper(X2), proper(X3)) | U33#(ok(X1), ok(X2), ok(X3)) | → | U33#(X1, X2, X3) | |
| active#(U14(X1, X2, X3)) | → | U14#(active(X1), X2, X3) | proper#(U12(X1, X2, X3)) | → | proper#(X2) | |
| proper#(U12(X1, X2, X3)) | → | U12#(proper(X1), proper(X2), proper(X3)) | active#(U81(tt, M, N)) | → | U82#(isNatKind(M), M, N) | |
| proper#(U72(X1, X2)) | → | proper#(X1) | proper#(U31(X1, X2, X3)) | → | proper#(X1) | |
| active#(s(X)) | → | s#(active(X)) | s#(ok(X)) | → | s#(X) | |
| U81#(ok(X1), ok(X2), ok(X3)) | → | U81#(X1, X2, X3) | proper#(U11(X1, X2, X3)) | → | proper#(X1) | |
| proper#(U14(X1, X2, X3)) | → | proper#(X3) | proper#(U14(X1, X2, X3)) | → | proper#(X2) | |
| active#(plus(N, s(M))) | → | isNat#(M) | active#(U72(X1, X2)) | → | U72#(active(X1), X2) | |
| proper#(s(X)) | → | s#(proper(X)) | active#(U23(X)) | → | active#(X) | |
| active#(x(X1, X2)) | → | active#(X2) | U36#(ok(X)) | → | U36#(X) | |
| active#(U61(tt, V2)) | → | U62#(isNatKind(V2)) | top#(ok(X)) | → | top#(active(X)) | |
| U92#(mark(X)) | → | U92#(X) | U23#(mark(X)) | → | U23#(X) | |
| U102#(ok(X1), ok(X2), ok(X3)) | → | U102#(X1, X2, X3) | proper#(U83(X1, X2, X3)) | → | proper#(X1) | |
| U42#(mark(X)) | → | U42#(X) | active#(U61(X1, X2)) | → | U61#(active(X1), X2) | |
| active#(U13(tt, V1, V2)) | → | isNatKind#(V2) | U82#(ok(X1), ok(X2), ok(X3)) | → | U82#(X1, X2, X3) | |
| active#(U81(tt, M, N)) | → | isNatKind#(M) | active#(U36(X)) | → | active#(X) | |
| plus#(X1, mark(X2)) | → | plus#(X1, X2) | proper#(U83(X1, X2, X3)) | → | proper#(X2) | |
| proper#(U36(X)) | → | U36#(proper(X)) | active#(U23(X)) | → | U23#(active(X)) | |
| U81#(mark(X1), X2, X3) | → | U81#(X1, X2, X3) | U84#(mark(X1), X2, X3) | → | U84#(X1, X2, X3) | |
| proper#(plus(X1, X2)) | → | plus#(proper(X1), proper(X2)) | active#(U62(X)) | → | U62#(active(X)) | |
| U32#(ok(X1), ok(X2), ok(X3)) | → | U32#(X1, X2, X3) | top#(mark(X)) | → | top#(proper(X)) | |
| active#(U31(tt, V1, V2)) | → | U32#(isNatKind(V1), V1, V2) | active#(U84(X1, X2, X3)) | → | active#(X1) | |
| proper#(U51(X)) | → | U51#(proper(X)) | proper#(U14(X1, X2, X3)) | → | U14#(proper(X1), proper(X2), proper(X3)) | |
| proper#(U22(X1, X2)) | → | proper#(X2) | U62#(ok(X)) | → | U62#(X) | |
| proper#(U102(X1, X2, X3)) | → | proper#(X3) | proper#(U84(X1, X2, X3)) | → | U84#(proper(X1), proper(X2), proper(X3)) | |
| active#(U16(X)) | → | active#(X) | active#(U82(tt, M, N)) | → | isNat#(N) | |
| active#(U11(X1, X2, X3)) | → | active#(X1) | proper#(U101(X1, X2, X3)) | → | U101#(proper(X1), proper(X2), proper(X3)) | |
| active#(U32(tt, V1, V2)) | → | U33#(isNatKind(V2), V1, V2) | U12#(ok(X1), ok(X2), ok(X3)) | → | U12#(X1, X2, X3) | |
| proper#(U82(X1, X2, X3)) | → | U82#(proper(X1), proper(X2), proper(X3)) | proper#(U32(X1, X2, X3)) | → | proper#(X1) | |
| active#(U34(tt, V1, V2)) | → | isNat#(V1) | active#(U71(tt, N)) | → | isNatKind#(N) | |
| active#(U81(X1, X2, X3)) | → | U81#(active(X1), X2, X3) | proper#(U82(X1, X2, X3)) | → | proper#(X3) | |
| proper#(U61(X1, X2)) | → | U61#(proper(X1), proper(X2)) | active#(U84(X1, X2, X3)) | → | U84#(active(X1), X2, X3) | |
| proper#(U102(X1, X2, X3)) | → | proper#(X1) | U92#(ok(X)) | → | U92#(X) | |
| active#(U71(X1, X2)) | → | U71#(active(X1), X2) | active#(U42(X)) | → | active#(X) | |
| proper#(U83(X1, X2, X3)) | → | U83#(proper(X1), proper(X2), proper(X3)) | proper#(x(X1, X2)) | → | proper#(X1) | |
| U101#(mark(X1), X2, X3) | → | U101#(X1, X2, X3) | proper#(U21(X1, X2)) | → | proper#(X2) | |
| proper#(U22(X1, X2)) | → | proper#(X1) | active#(plus(N, 0)) | → | U71#(isNat(N), N) | |
| proper#(U71(X1, X2)) | → | proper#(X1) | proper#(U31(X1, X2, X3)) | → | U31#(proper(X1), proper(X2), proper(X3)) | |
| proper#(U102(X1, X2, X3)) | → | proper#(X2) | active#(isNatKind(x(V1, V2))) | → | isNatKind#(V1) | |
| U23#(ok(X)) | → | U23#(X) | active#(U14(tt, V1, V2)) | → | isNat#(V1) | |
| proper#(x(X1, X2)) | → | x#(proper(X1), proper(X2)) | proper#(U31(X1, X2, X3)) | → | proper#(X3) | |
| active#(isNatKind(s(V1))) | → | isNatKind#(V1) | proper#(U21(X1, X2)) | → | U21#(proper(X1), proper(X2)) | |
| proper#(U34(X1, X2, X3)) | → | proper#(X3) | U35#(mark(X1), X2) | → | U35#(X1, X2) | |
| active#(x(X1, X2)) | → | active#(X1) | active#(U91(tt, N)) | → | U92#(isNatKind(N)) | |
| active#(U11(tt, V1, V2)) | → | U12#(isNatKind(V1), V1, V2) | active#(U41(tt, V2)) | → | isNatKind#(V2) | |
| proper#(U61(X1, X2)) | → | proper#(X1) | active#(plus(X1, X2)) | → | plus#(X1, active(X2)) | |
| U16#(ok(X)) | → | U16#(X) | active#(U21(X1, X2)) | → | active#(X1) | |
| active#(U83(tt, M, N)) | → | U84#(isNatKind(N), M, N) | proper#(U82(X1, X2, X3)) | → | proper#(X2) | |
| active#(plus(X1, X2)) | → | plus#(active(X1), X2) | U21#(mark(X1), X2) | → | U21#(X1, X2) | |
| active#(U51(X)) | → | active#(X) | U31#(mark(X1), X2, X3) | → | U31#(X1, X2, X3) | |
| proper#(U36(X)) | → | proper#(X) | proper#(U35(X1, X2)) | → | U35#(proper(X1), proper(X2)) | |
| proper#(U83(X1, X2, X3)) | → | proper#(X3) |
| active(U101(tt, M, N)) | → | mark(U102(isNatKind(M), M, N)) | active(U102(tt, M, N)) | → | mark(U103(isNat(N), M, N)) | |
| active(U103(tt, M, N)) | → | mark(U104(isNatKind(N), M, N)) | active(U104(tt, M, N)) | → | mark(plus(x(N, M), N)) | |
| active(U11(tt, V1, V2)) | → | mark(U12(isNatKind(V1), V1, V2)) | active(U12(tt, V1, V2)) | → | mark(U13(isNatKind(V2), V1, V2)) | |
| active(U13(tt, V1, V2)) | → | mark(U14(isNatKind(V2), V1, V2)) | active(U14(tt, V1, V2)) | → | mark(U15(isNat(V1), V2)) | |
| active(U15(tt, V2)) | → | mark(U16(isNat(V2))) | active(U16(tt)) | → | mark(tt) | |
| active(U21(tt, V1)) | → | mark(U22(isNatKind(V1), V1)) | active(U22(tt, V1)) | → | mark(U23(isNat(V1))) | |
| active(U23(tt)) | → | mark(tt) | active(U31(tt, V1, V2)) | → | mark(U32(isNatKind(V1), V1, V2)) | |
| active(U32(tt, V1, V2)) | → | mark(U33(isNatKind(V2), V1, V2)) | active(U33(tt, V1, V2)) | → | mark(U34(isNatKind(V2), V1, V2)) | |
| active(U34(tt, V1, V2)) | → | mark(U35(isNat(V1), V2)) | active(U35(tt, V2)) | → | mark(U36(isNat(V2))) | |
| active(U36(tt)) | → | mark(tt) | active(U41(tt, V2)) | → | mark(U42(isNatKind(V2))) | |
| active(U42(tt)) | → | mark(tt) | active(U51(tt)) | → | mark(tt) | |
| active(U61(tt, V2)) | → | mark(U62(isNatKind(V2))) | active(U62(tt)) | → | mark(tt) | |
| active(U71(tt, N)) | → | mark(U72(isNatKind(N), N)) | active(U72(tt, N)) | → | mark(N) | |
| active(U81(tt, M, N)) | → | mark(U82(isNatKind(M), M, N)) | active(U82(tt, M, N)) | → | mark(U83(isNat(N), M, N)) | |
| active(U83(tt, M, N)) | → | mark(U84(isNatKind(N), M, N)) | active(U84(tt, M, N)) | → | mark(s(plus(N, M))) | |
| active(U91(tt, N)) | → | mark(U92(isNatKind(N))) | active(U92(tt)) | → | mark(0) | |
| active(isNat(0)) | → | mark(tt) | active(isNat(plus(V1, V2))) | → | mark(U11(isNatKind(V1), V1, V2)) | |
| active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | active(isNat(x(V1, V2))) | → | mark(U31(isNatKind(V1), V1, V2)) | |
| active(isNatKind(0)) | → | mark(tt) | active(isNatKind(plus(V1, V2))) | → | mark(U41(isNatKind(V1), V2)) | |
| active(isNatKind(s(V1))) | → | mark(U51(isNatKind(V1))) | active(isNatKind(x(V1, V2))) | → | mark(U61(isNatKind(V1), V2)) | |
| active(plus(N, 0)) | → | mark(U71(isNat(N), N)) | active(plus(N, s(M))) | → | mark(U81(isNat(M), M, N)) | |
| active(x(N, 0)) | → | mark(U91(isNat(N), N)) | active(x(N, s(M))) | → | mark(U101(isNat(M), M, N)) | |
| active(U101(X1, X2, X3)) | → | U101(active(X1), X2, X3) | active(U102(X1, X2, X3)) | → | U102(active(X1), X2, X3) | |
| active(U103(X1, X2, X3)) | → | U103(active(X1), X2, X3) | active(U104(X1, X2, X3)) | → | U104(active(X1), X2, X3) | |
| active(plus(X1, X2)) | → | plus(active(X1), X2) | active(plus(X1, X2)) | → | plus(X1, active(X2)) | |
| active(x(X1, X2)) | → | x(active(X1), X2) | active(x(X1, X2)) | → | x(X1, active(X2)) | |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | active(U12(X1, X2, X3)) | → | U12(active(X1), X2, X3) | |
| active(U13(X1, X2, X3)) | → | U13(active(X1), X2, X3) | active(U14(X1, X2, X3)) | → | U14(active(X1), X2, X3) | |
| active(U15(X1, X2)) | → | U15(active(X1), X2) | active(U16(X)) | → | U16(active(X)) | |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | active(U22(X1, X2)) | → | U22(active(X1), X2) | |
| active(U23(X)) | → | U23(active(X)) | active(U31(X1, X2, X3)) | → | U31(active(X1), X2, X3) | |
| active(U32(X1, X2, X3)) | → | U32(active(X1), X2, X3) | active(U33(X1, X2, X3)) | → | U33(active(X1), X2, X3) | |
| active(U34(X1, X2, X3)) | → | U34(active(X1), X2, X3) | active(U35(X1, X2)) | → | U35(active(X1), X2) | |
| active(U36(X)) | → | U36(active(X)) | active(U41(X1, X2)) | → | U41(active(X1), X2) | |
| active(U42(X)) | → | U42(active(X)) | active(U51(X)) | → | U51(active(X)) | |
| active(U61(X1, X2)) | → | U61(active(X1), X2) | active(U62(X)) | → | U62(active(X)) | |
| active(U71(X1, X2)) | → | U71(active(X1), X2) | active(U72(X1, X2)) | → | U72(active(X1), X2) | |
| active(U81(X1, X2, X3)) | → | U81(active(X1), X2, X3) | active(U82(X1, X2, X3)) | → | U82(active(X1), X2, X3) | |
| active(U83(X1, X2, X3)) | → | U83(active(X1), X2, X3) | active(U84(X1, X2, X3)) | → | U84(active(X1), X2, X3) | |
| active(s(X)) | → | s(active(X)) | active(U91(X1, X2)) | → | U91(active(X1), X2) | |
| active(U92(X)) | → | U92(active(X)) | U101(mark(X1), X2, X3) | → | mark(U101(X1, X2, X3)) | |
| U102(mark(X1), X2, X3) | → | mark(U102(X1, X2, X3)) | U103(mark(X1), X2, X3) | → | mark(U103(X1, X2, X3)) | |
| U104(mark(X1), X2, X3) | → | mark(U104(X1, X2, X3)) | plus(mark(X1), X2) | → | mark(plus(X1, X2)) | |
| plus(X1, mark(X2)) | → | mark(plus(X1, X2)) | x(mark(X1), X2) | → | mark(x(X1, X2)) | |
| x(X1, mark(X2)) | → | mark(x(X1, X2)) | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) | |
| U12(mark(X1), X2, X3) | → | mark(U12(X1, X2, X3)) | U13(mark(X1), X2, X3) | → | mark(U13(X1, X2, X3)) | |
| U14(mark(X1), X2, X3) | → | mark(U14(X1, X2, X3)) | U15(mark(X1), X2) | → | mark(U15(X1, X2)) | |
| U16(mark(X)) | → | mark(U16(X)) | U21(mark(X1), X2) | → | mark(U21(X1, X2)) | |
| U22(mark(X1), X2) | → | mark(U22(X1, X2)) | U23(mark(X)) | → | mark(U23(X)) | |
| U31(mark(X1), X2, X3) | → | mark(U31(X1, X2, X3)) | U32(mark(X1), X2, X3) | → | mark(U32(X1, X2, X3)) | |
| U33(mark(X1), X2, X3) | → | mark(U33(X1, X2, X3)) | U34(mark(X1), X2, X3) | → | mark(U34(X1, X2, X3)) | |
| U35(mark(X1), X2) | → | mark(U35(X1, X2)) | U36(mark(X)) | → | mark(U36(X)) | |
| U41(mark(X1), X2) | → | mark(U41(X1, X2)) | U42(mark(X)) | → | mark(U42(X)) | |
| U51(mark(X)) | → | mark(U51(X)) | U61(mark(X1), X2) | → | mark(U61(X1, X2)) | |
| U62(mark(X)) | → | mark(U62(X)) | U71(mark(X1), X2) | → | mark(U71(X1, X2)) | |
| U72(mark(X1), X2) | → | mark(U72(X1, X2)) | U81(mark(X1), X2, X3) | → | mark(U81(X1, X2, X3)) | |
| U82(mark(X1), X2, X3) | → | mark(U82(X1, X2, X3)) | U83(mark(X1), X2, X3) | → | mark(U83(X1, X2, X3)) | |
| U84(mark(X1), X2, X3) | → | mark(U84(X1, X2, X3)) | s(mark(X)) | → | mark(s(X)) | |
| U91(mark(X1), X2) | → | mark(U91(X1, X2)) | U92(mark(X)) | → | mark(U92(X)) | |
| proper(U101(X1, X2, X3)) | → | U101(proper(X1), proper(X2), proper(X3)) | proper(tt) | → | ok(tt) | |
| proper(U102(X1, X2, X3)) | → | U102(proper(X1), proper(X2), proper(X3)) | proper(isNatKind(X)) | → | isNatKind(proper(X)) | |
| proper(U103(X1, X2, X3)) | → | U103(proper(X1), proper(X2), proper(X3)) | proper(isNat(X)) | → | isNat(proper(X)) | |
| proper(U104(X1, X2, X3)) | → | U104(proper(X1), proper(X2), proper(X3)) | proper(plus(X1, X2)) | → | plus(proper(X1), proper(X2)) | |
| proper(x(X1, X2)) | → | x(proper(X1), proper(X2)) | proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | |
| proper(U12(X1, X2, X3)) | → | U12(proper(X1), proper(X2), proper(X3)) | proper(U13(X1, X2, X3)) | → | U13(proper(X1), proper(X2), proper(X3)) | |
| proper(U14(X1, X2, X3)) | → | U14(proper(X1), proper(X2), proper(X3)) | proper(U15(X1, X2)) | → | U15(proper(X1), proper(X2)) | |
| proper(U16(X)) | → | U16(proper(X)) | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | |
| proper(U22(X1, X2)) | → | U22(proper(X1), proper(X2)) | proper(U23(X)) | → | U23(proper(X)) | |
| proper(U31(X1, X2, X3)) | → | U31(proper(X1), proper(X2), proper(X3)) | proper(U32(X1, X2, X3)) | → | U32(proper(X1), proper(X2), proper(X3)) | |
| proper(U33(X1, X2, X3)) | → | U33(proper(X1), proper(X2), proper(X3)) | proper(U34(X1, X2, X3)) | → | U34(proper(X1), proper(X2), proper(X3)) | |
| proper(U35(X1, X2)) | → | U35(proper(X1), proper(X2)) | proper(U36(X)) | → | U36(proper(X)) | |
| proper(U41(X1, X2)) | → | U41(proper(X1), proper(X2)) | proper(U42(X)) | → | U42(proper(X)) | |
| proper(U51(X)) | → | U51(proper(X)) | proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | |
| proper(U62(X)) | → | U62(proper(X)) | proper(U71(X1, X2)) | → | U71(proper(X1), proper(X2)) | |
| proper(U72(X1, X2)) | → | U72(proper(X1), proper(X2)) | proper(U81(X1, X2, X3)) | → | U81(proper(X1), proper(X2), proper(X3)) | |
| proper(U82(X1, X2, X3)) | → | U82(proper(X1), proper(X2), proper(X3)) | proper(U83(X1, X2, X3)) | → | U83(proper(X1), proper(X2), proper(X3)) | |
| proper(U84(X1, X2, X3)) | → | U84(proper(X1), proper(X2), proper(X3)) | proper(s(X)) | → | s(proper(X)) | |
| proper(U91(X1, X2)) | → | U91(proper(X1), proper(X2)) | proper(U92(X)) | → | U92(proper(X)) | |
| proper(0) | → | ok(0) | U101(ok(X1), ok(X2), ok(X3)) | → | ok(U101(X1, X2, X3)) | |
| U102(ok(X1), ok(X2), ok(X3)) | → | ok(U102(X1, X2, X3)) | isNatKind(ok(X)) | → | ok(isNatKind(X)) | |
| U103(ok(X1), ok(X2), ok(X3)) | → | ok(U103(X1, X2, X3)) | isNat(ok(X)) | → | ok(isNat(X)) | |
| U104(ok(X1), ok(X2), ok(X3)) | → | ok(U104(X1, X2, X3)) | plus(ok(X1), ok(X2)) | → | ok(plus(X1, X2)) | |
| x(ok(X1), ok(X2)) | → | ok(x(X1, X2)) | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | |
| U12(ok(X1), ok(X2), ok(X3)) | → | ok(U12(X1, X2, X3)) | U13(ok(X1), ok(X2), ok(X3)) | → | ok(U13(X1, X2, X3)) | |
| U14(ok(X1), ok(X2), ok(X3)) | → | ok(U14(X1, X2, X3)) | U15(ok(X1), ok(X2)) | → | ok(U15(X1, X2)) | |
| U16(ok(X)) | → | ok(U16(X)) | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | |
| U22(ok(X1), ok(X2)) | → | ok(U22(X1, X2)) | U23(ok(X)) | → | ok(U23(X)) | |
| U31(ok(X1), ok(X2), ok(X3)) | → | ok(U31(X1, X2, X3)) | U32(ok(X1), ok(X2), ok(X3)) | → | ok(U32(X1, X2, X3)) | |
| U33(ok(X1), ok(X2), ok(X3)) | → | ok(U33(X1, X2, X3)) | U34(ok(X1), ok(X2), ok(X3)) | → | ok(U34(X1, X2, X3)) | |
| U35(ok(X1), ok(X2)) | → | ok(U35(X1, X2)) | U36(ok(X)) | → | ok(U36(X)) | |
| U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | U42(ok(X)) | → | ok(U42(X)) | |
| U51(ok(X)) | → | ok(U51(X)) | U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) | |
| U62(ok(X)) | → | ok(U62(X)) | U71(ok(X1), ok(X2)) | → | ok(U71(X1, X2)) | |
| U72(ok(X1), ok(X2)) | → | ok(U72(X1, X2)) | U81(ok(X1), ok(X2), ok(X3)) | → | ok(U81(X1, X2, X3)) | |
| U82(ok(X1), ok(X2), ok(X3)) | → | ok(U82(X1, X2, X3)) | U83(ok(X1), ok(X2), ok(X3)) | → | ok(U83(X1, X2, X3)) | |
| U84(ok(X1), ok(X2), ok(X3)) | → | ok(U84(X1, X2, X3)) | s(ok(X)) | → | ok(s(X)) | |
| U91(ok(X1), ok(X2)) | → | ok(U91(X1, X2)) | U92(ok(X)) | → | ok(U92(X)) | |
| top(mark(X)) | → | top(proper(X)) | top(ok(X)) | → | top(active(X)) |
Termination of terms over the following signature is verified: U104, isNat, U62, U61, U42, U92, U41, U91, ok, proper, U23, U21, top, U22, plus, isNatKind, U83, U84, mark, U71, U36, U72, 0, s, U51, U14, tt, U15, U82, U16, U81, U11, active, U12, U13, U31, U102, U32, U103, U33, U34, x, U101, U35