The TRS could not be proven terminating. The proof attempt took 60001 ms.
Problem 1 remains open; application of the following processors failed [DependencyGraph (timeout)].
| U12#(active(X1), X2) | → | U12#(X1, X2) | mark#(U61(X)) | → | U61#(mark(X)) | |
| mark#(U11(X1, X2, X3)) | → | U11#(mark(X1), X2, X3) | active#(U71(tt, M, N)) | → | x#(N, M) | |
| U31#(X1, active(X2), X3) | → | U31#(X1, X2, X3) | x#(active(X1), X2) | → | x#(X1, X2) | |
| x#(X1, active(X2)) | → | x#(X1, X2) | active#(plus(N, 0)) | → | and#(isNat(N), isNatKind(N)) | |
| active#(isNatKind(x(V1, V2))) | → | isNatKind#(V2) | active#(U61(tt)) | → | mark#(0) | |
| isNat#(active(X)) | → | isNat#(X) | U21#(X1, active(X2)) | → | U21#(X1, X2) | |
| mark#(s(X)) | → | mark#(X) | U71#(X1, mark(X2), X3) | → | U71#(X1, X2, X3) | |
| mark#(U12(X1, X2)) | → | mark#(X1) | active#(isNat(x(V1, V2))) | → | mark#(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | |
| mark#(U31(X1, X2, X3)) | → | active#(U31(mark(X1), X2, X3)) | mark#(plus(X1, X2)) | → | active#(plus(mark(X1), mark(X2))) | |
| active#(plus(N, s(M))) | → | and#(isNat(N), isNatKind(N)) | active#(U11(tt, V1, V2)) | → | mark#(U12(isNat(V1), V2)) | |
| and#(mark(X1), X2) | → | and#(X1, X2) | active#(U21(tt, V1)) | → | isNat#(V1) | |
| active#(plus(N, s(M))) | → | isNatKind#(M) | active#(U51(tt, M, N)) | → | s#(plus(N, M)) | |
| mark#(U22(X)) | → | U22#(mark(X)) | mark#(U22(X)) | → | active#(U22(mark(X))) | |
| mark#(U11(X1, X2, X3)) | → | mark#(X1) | mark#(U13(X)) | → | U13#(mark(X)) | |
| U12#(X1, active(X2)) | → | U12#(X1, X2) | active#(x(N, s(M))) | → | and#(isNat(M), isNatKind(M)) | |
| U31#(active(X1), X2, X3) | → | U31#(X1, X2, X3) | active#(U71(tt, M, N)) | → | plus#(x(N, M), N) | |
| active#(isNat(x(V1, V2))) | → | isNatKind#(V1) | mark#(U21(X1, X2)) | → | active#(U21(mark(X1), X2)) | |
| U32#(X1, active(X2)) | → | U32#(X1, X2) | U11#(active(X1), X2, X3) | → | U11#(X1, X2, X3) | |
| mark#(U33(X)) | → | active#(U33(mark(X))) | U71#(active(X1), X2, X3) | → | U71#(X1, X2, X3) | |
| active#(U32(tt, V2)) | → | isNat#(V2) | mark#(U41(X1, X2)) | → | mark#(X1) | |
| U41#(X1, active(X2)) | → | U41#(X1, X2) | mark#(U61(X)) | → | mark#(X) | |
| isNatKind#(mark(X)) | → | isNatKind#(X) | mark#(isNatKind(X)) | → | isNatKind#(X) | |
| active#(U11(tt, V1, V2)) | → | U12#(isNat(V1), V2) | mark#(x(X1, X2)) | → | active#(x(mark(X1), mark(X2))) | |
| U21#(active(X1), X2) | → | U21#(X1, X2) | U33#(mark(X)) | → | U33#(X) | |
| active#(U31(tt, V1, V2)) | → | isNat#(V1) | active#(plus(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | |
| mark#(U71(X1, X2, X3)) | → | U71#(mark(X1), X2, X3) | U31#(X1, X2, active(X3)) | → | U31#(X1, X2, X3) | |
| active#(x(N, s(M))) | → | isNat#(M) | active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | |
| active#(x(N, s(M))) | → | isNat#(N) | U33#(active(X)) | → | U33#(X) | |
| mark#(x(X1, X2)) | → | mark#(X1) | active#(plus(N, s(M))) | → | and#(isNat(M), isNatKind(M)) | |
| active#(isNatKind(x(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | active#(isNat(x(V1, V2))) | → | isNatKind#(V2) | |
| active#(plus(N, s(M))) | → | mark#(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | mark#(s(X)) | → | s#(mark(X)) | |
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | U12#(mark(X1), X2) | → | U12#(X1, X2) | |
| U22#(mark(X)) | → | U22#(X) | active#(isNatKind(plus(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | |
| active#(x(N, s(M))) | → | and#(isNat(N), isNatKind(N)) | active#(U32(tt, V2)) | → | mark#(U33(isNat(V2))) | |
| U51#(X1, mark(X2), X3) | → | U51#(X1, X2, X3) | U41#(X1, mark(X2)) | → | U41#(X1, X2) | |
| active#(isNat(0)) | → | mark#(tt) | mark#(and(X1, X2)) | → | and#(mark(X1), X2) | |
| active#(isNatKind(x(V1, V2))) | → | mark#(and(isNatKind(V1), isNatKind(V2))) | U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | |
| active#(U12(tt, V2)) | → | U13#(isNat(V2)) | mark#(U31(X1, X2, X3)) | → | mark#(X1) | |
| active#(U21(tt, V1)) | → | U22#(isNat(V1)) | mark#(plus(X1, X2)) | → | mark#(X1) | |
| mark#(x(X1, X2)) | → | mark#(X2) | isNatKind#(active(X)) | → | isNatKind#(X) | |
| U22#(active(X)) | → | U22#(X) | active#(x(N, 0)) | → | U61#(and(isNat(N), isNatKind(N))) | |
| U51#(active(X1), X2, X3) | → | U51#(X1, X2, X3) | active#(plus(N, s(M))) | → | isNat#(N) | |
| and#(X1, active(X2)) | → | and#(X1, X2) | active#(plus(N, s(M))) | → | isNatKind#(N) | |
| U71#(X1, X2, active(X3)) | → | U71#(X1, X2, X3) | mark#(U21(X1, X2)) | → | mark#(X1) | |
| mark#(U21(X1, X2)) | → | U21#(mark(X1), X2) | mark#(isNat(X)) | → | isNat#(X) | |
| isNat#(mark(X)) | → | isNat#(X) | active#(isNat(plus(V1, V2))) | → | isNatKind#(V1) | |
| active#(isNat(plus(V1, V2))) | → | isNatKind#(V2) | U11#(X1, X2, mark(X3)) | → | U11#(X1, X2, X3) | |
| mark#(U61(X)) | → | active#(U61(mark(X))) | U31#(X1, X2, mark(X3)) | → | U31#(X1, X2, X3) | |
| U41#(mark(X1), X2) | → | U41#(X1, X2) | active#(U13(tt)) | → | mark#(tt) | |
| active#(isNat(plus(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | and#(X1, mark(X2)) | → | and#(X1, X2) | |
| U12#(X1, mark(X2)) | → | U12#(X1, X2) | mark#(U32(X1, X2)) | → | active#(U32(mark(X1), X2)) | |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V2) | active#(isNat(s(V1))) | → | mark#(U21(isNatKind(V1), V1)) | |
| mark#(plus(X1, X2)) | → | plus#(mark(X1), mark(X2)) | mark#(U12(X1, X2)) | → | U12#(mark(X1), X2) | |
| s#(mark(X)) | → | s#(X) | U32#(active(X1), X2) | → | U32#(X1, X2) | |
| mark#(U51(X1, X2, X3)) | → | U51#(mark(X1), X2, X3) | active#(isNat(plus(V1, V2))) | → | mark#(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | |
| U31#(X1, mark(X2), X3) | → | U31#(X1, X2, X3) | mark#(U13(X)) | → | mark#(X) | |
| active#(isNatKind(0)) | → | mark#(tt) | active#(isNat(x(V1, V2))) | → | and#(isNatKind(V1), isNatKind(V2)) | |
| active#(U32(tt, V2)) | → | U33#(isNat(V2)) | active#(isNat(plus(V1, V2))) | → | U11#(and(isNatKind(V1), isNatKind(V2)), V1, V2) | |
| active#(isNatKind(plus(V1, V2))) | → | isNatKind#(V1) | active#(x(N, s(M))) | → | isNatKind#(N) | |
| active#(U22(tt)) | → | mark#(tt) | active#(U12(tt, V2)) | → | isNat#(V2) | |
| x#(X1, mark(X2)) | → | x#(X1, X2) | mark#(U31(X1, X2, X3)) | → | U31#(mark(X1), X2, X3) | |
| active#(plus(N, 0)) | → | isNat#(N) | U51#(X1, X2, active(X3)) | → | U51#(X1, X2, X3) | |
| U11#(X1, active(X2), X3) | → | U11#(X1, X2, X3) | U71#(mark(X1), X2, X3) | → | U71#(X1, X2, X3) | |
| U41#(active(X1), X2) | → | U41#(X1, X2) | mark#(U33(X)) | → | U33#(mark(X)) | |
| active#(isNat(s(V1))) | → | isNatKind#(V1) | mark#(and(X1, X2)) | → | mark#(X1) | |
| mark#(U32(X1, X2)) | → | mark#(X1) | U61#(active(X)) | → | U61#(X) | |
| active#(U12(tt, V2)) | → | mark#(U13(isNat(V2))) | plus#(mark(X1), X2) | → | plus#(X1, X2) | |
| active#(x(N, 0)) | → | mark#(U61(and(isNat(N), isNatKind(N)))) | plus#(active(X1), X2) | → | plus#(X1, X2) | |
| active#(plus(N, 0)) | → | mark#(U41(and(isNat(N), isNatKind(N)), N)) | active#(x(N, 0)) | → | isNat#(N) | |
| mark#(tt) | → | active#(tt) | active#(plus(N, 0)) | → | U41#(and(isNat(N), isNatKind(N)), N) | |
| mark#(U33(X)) | → | mark#(X) | active#(x(N, s(M))) | → | isNatKind#(M) | |
| active#(U11(tt, V1, V2)) | → | isNat#(V1) | mark#(U51(X1, X2, X3)) | → | mark#(X1) | |
| U21#(X1, mark(X2)) | → | U21#(X1, X2) | x#(mark(X1), X2) | → | x#(X1, X2) | |
| active#(plus(N, s(M))) | → | U51#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | U13#(mark(X)) | → | U13#(X) | |
| mark#(U13(X)) | → | active#(U13(mark(X))) | active#(isNatKind(s(V1))) | → | mark#(isNatKind(V1)) | |
| active#(U41(tt, N)) | → | mark#(N) | active#(x(N, 0)) | → | isNatKind#(N) | |
| mark#(0) | → | active#(0) | U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | |
| mark#(s(X)) | → | active#(s(mark(X))) | active#(U51(tt, M, N)) | → | mark#(s(plus(N, M))) | |
| U61#(mark(X)) | → | U61#(X) | active#(isNat(x(V1, V2))) | → | U31#(and(isNatKind(V1), isNatKind(V2)), V1, V2) | |
| U71#(X1, X2, mark(X3)) | → | U71#(X1, X2, X3) | mark#(U22(X)) | → | mark#(X) | |
| active#(U51(tt, M, N)) | → | plus#(N, M) | active#(plus(N, s(M))) | → | isNat#(M) | |
| mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | mark#(U12(X1, X2)) | → | active#(U12(mark(X1), X2)) | |
| active#(x(N, 0)) | → | and#(isNat(N), isNatKind(N)) | U13#(active(X)) | → | U13#(X) | |
| mark#(x(X1, X2)) | → | x#(mark(X1), mark(X2)) | active#(U31(tt, V1, V2)) | → | U32#(isNat(V1), V2) | |
| U11#(X1, X2, active(X3)) | → | U11#(X1, X2, X3) | active#(x(N, s(M))) | → | U71#(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | |
| active#(U21(tt, V1)) | → | mark#(U22(isNat(V1))) | mark#(U71(X1, X2, X3)) | → | active#(U71(mark(X1), X2, X3)) | |
| mark#(isNatKind(X)) | → | active#(isNatKind(X)) | active#(x(N, s(M))) | → | mark#(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | U51#(X1, active(X2), X3) | → | U51#(X1, X2, X3) | |
| active#(U33(tt)) | → | mark#(tt) | plus#(X1, mark(X2)) | → | plus#(X1, X2) | |
| U71#(X1, active(X2), X3) | → | U71#(X1, X2, X3) | mark#(U71(X1, X2, X3)) | → | mark#(X1) | |
| mark#(U32(X1, X2)) | → | U32#(mark(X1), X2) | active#(isNatKind(plus(V1, V2))) | → | mark#(and(isNatKind(V1), isNatKind(V2))) | |
| active#(U31(tt, V1, V2)) | → | mark#(U32(isNat(V1), V2)) | mark#(U41(X1, X2)) | → | U41#(mark(X1), X2) | |
| plus#(X1, active(X2)) | → | plus#(X1, X2) | and#(active(X1), X2) | → | and#(X1, X2) | |
| mark#(isNat(X)) | → | active#(isNat(X)) | active#(plus(N, 0)) | → | isNatKind#(N) | |
| U32#(X1, mark(X2)) | → | U32#(X1, X2) | mark#(plus(X1, X2)) | → | mark#(X2) | |
| mark#(and(X1, X2)) | → | active#(and(mark(X1), X2)) | active#(isNatKind(x(V1, V2))) | → | isNatKind#(V1) | |
| active#(x(N, s(M))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | active#(isNatKind(s(V1))) | → | isNatKind#(V1) | |
| active#(U71(tt, M, N)) | → | mark#(plus(x(N, M), N)) | U51#(X1, X2, mark(X3)) | → | U51#(X1, X2, X3) | |
| active#(and(tt, X)) | → | mark#(X) | U21#(mark(X1), X2) | → | U21#(X1, X2) | |
| U32#(mark(X1), X2) | → | U32#(X1, X2) | U11#(X1, mark(X2), X3) | → | U11#(X1, X2, X3) | |
| s#(active(X)) | → | s#(X) | U31#(mark(X1), X2, X3) | → | U31#(X1, X2, X3) |
| active(U11(tt, V1, V2)) | → | mark(U12(isNat(V1), V2)) | active(U12(tt, V2)) | → | mark(U13(isNat(V2))) | |
| active(U13(tt)) | → | mark(tt) | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) | |
| active(U22(tt)) | → | mark(tt) | active(U31(tt, V1, V2)) | → | mark(U32(isNat(V1), V2)) | |
| active(U32(tt, V2)) | → | mark(U33(isNat(V2))) | active(U33(tt)) | → | mark(tt) | |
| active(U41(tt, N)) | → | mark(N) | active(U51(tt, M, N)) | → | mark(s(plus(N, M))) | |
| active(U61(tt)) | → | mark(0) | active(U71(tt, M, N)) | → | mark(plus(x(N, M), N)) | |
| active(and(tt, X)) | → | mark(X) | active(isNat(0)) | → | mark(tt) | |
| active(isNat(plus(V1, V2))) | → | mark(U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | |
| active(isNat(x(V1, V2))) | → | mark(U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)) | active(isNatKind(0)) | → | mark(tt) | |
| active(isNatKind(plus(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | |
| active(isNatKind(x(V1, V2))) | → | mark(and(isNatKind(V1), isNatKind(V2))) | active(plus(N, 0)) | → | mark(U41(and(isNat(N), isNatKind(N)), N)) | |
| active(plus(N, s(M))) | → | mark(U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | active(x(N, 0)) | → | mark(U61(and(isNat(N), isNatKind(N)))) | |
| active(x(N, s(M))) | → | mark(U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)) | mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | |
| mark(tt) | → | active(tt) | mark(U12(X1, X2)) | → | active(U12(mark(X1), X2)) | |
| mark(isNat(X)) | → | active(isNat(X)) | mark(U13(X)) | → | active(U13(mark(X))) | |
| mark(U21(X1, X2)) | → | active(U21(mark(X1), X2)) | mark(U22(X)) | → | active(U22(mark(X))) | |
| mark(U31(X1, X2, X3)) | → | active(U31(mark(X1), X2, X3)) | mark(U32(X1, X2)) | → | active(U32(mark(X1), X2)) | |
| mark(U33(X)) | → | active(U33(mark(X))) | mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | |
| mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) | mark(s(X)) | → | active(s(mark(X))) | |
| mark(plus(X1, X2)) | → | active(plus(mark(X1), mark(X2))) | mark(U61(X)) | → | active(U61(mark(X))) | |
| mark(0) | → | active(0) | mark(U71(X1, X2, X3)) | → | active(U71(mark(X1), X2, X3)) | |
| mark(x(X1, X2)) | → | active(x(mark(X1), mark(X2))) | mark(and(X1, X2)) | → | active(and(mark(X1), X2)) | |
| mark(isNatKind(X)) | → | active(isNatKind(X)) | U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) | |
| U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | |
| U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | |
| U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | U12(mark(X1), X2) | → | U12(X1, X2) | |
| U12(X1, mark(X2)) | → | U12(X1, X2) | U12(active(X1), X2) | → | U12(X1, X2) | |
| U12(X1, active(X2)) | → | U12(X1, X2) | isNat(mark(X)) | → | isNat(X) | |
| isNat(active(X)) | → | isNat(X) | U13(mark(X)) | → | U13(X) | |
| U13(active(X)) | → | U13(X) | U21(mark(X1), X2) | → | U21(X1, X2) | |
| U21(X1, mark(X2)) | → | U21(X1, X2) | U21(active(X1), X2) | → | U21(X1, X2) | |
| U21(X1, active(X2)) | → | U21(X1, X2) | U22(mark(X)) | → | U22(X) | |
| U22(active(X)) | → | U22(X) | U31(mark(X1), X2, X3) | → | U31(X1, X2, X3) | |
| U31(X1, mark(X2), X3) | → | U31(X1, X2, X3) | U31(X1, X2, mark(X3)) | → | U31(X1, X2, X3) | |
| U31(active(X1), X2, X3) | → | U31(X1, X2, X3) | U31(X1, active(X2), X3) | → | U31(X1, X2, X3) | |
| U31(X1, X2, active(X3)) | → | U31(X1, X2, X3) | U32(mark(X1), X2) | → | U32(X1, X2) | |
| U32(X1, mark(X2)) | → | U32(X1, X2) | U32(active(X1), X2) | → | U32(X1, X2) | |
| U32(X1, active(X2)) | → | U32(X1, X2) | U33(mark(X)) | → | U33(X) | |
| U33(active(X)) | → | U33(X) | U41(mark(X1), X2) | → | U41(X1, X2) | |
| U41(X1, mark(X2)) | → | U41(X1, X2) | U41(active(X1), X2) | → | U41(X1, X2) | |
| U41(X1, active(X2)) | → | U41(X1, X2) | U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | |
| U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) | U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | |
| U51(active(X1), X2, X3) | → | U51(X1, X2, X3) | U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | |
| U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) | s(mark(X)) | → | s(X) | |
| s(active(X)) | → | s(X) | plus(mark(X1), X2) | → | plus(X1, X2) | |
| plus(X1, mark(X2)) | → | plus(X1, X2) | plus(active(X1), X2) | → | plus(X1, X2) | |
| plus(X1, active(X2)) | → | plus(X1, X2) | U61(mark(X)) | → | U61(X) | |
| U61(active(X)) | → | U61(X) | U71(mark(X1), X2, X3) | → | U71(X1, X2, X3) | |
| U71(X1, mark(X2), X3) | → | U71(X1, X2, X3) | U71(X1, X2, mark(X3)) | → | U71(X1, X2, X3) | |
| U71(active(X1), X2, X3) | → | U71(X1, X2, X3) | U71(X1, active(X2), X3) | → | U71(X1, X2, X3) | |
| U71(X1, X2, active(X3)) | → | U71(X1, X2, X3) | x(mark(X1), X2) | → | x(X1, X2) | |
| x(X1, mark(X2)) | → | x(X1, X2) | x(active(X1), X2) | → | x(X1, X2) | |
| x(X1, active(X2)) | → | x(X1, X2) | and(mark(X1), X2) | → | and(X1, X2) | |
| and(X1, mark(X2)) | → | and(X1, X2) | and(active(X1), X2) | → | and(X1, X2) | |
| and(X1, active(X2)) | → | and(X1, X2) | isNatKind(mark(X)) | → | isNatKind(X) | |
| isNatKind(active(X)) | → | isNatKind(X) |
Termination of terms over the following signature is verified: plus, isNatKind, mark, and, U71, isNat, U61, 0, U51, s, tt, U41, active, U11, U12, U13, U31, U32, U33, U21, x, U22