The TRS could not be proven terminating. The proof attempt took 60000 ms.
Problem 1 was processed with processor DependencyGraph (7088ms). | – Problem 2 remains open; application of the following processors failed [SubtermCriterion (2ms), DependencyGraph (4747ms), PolynomialLinearRange4iUR (timeout), DependencyGraph (4709ms), PolynomialLinearRange8NegiUR (timeout), DependencyGraph (timeout)].
| a__U11#(tt, V1, V2) | → | a__isNat#(V1) | a__U32#(tt, V2) | → | a__isNat#(V2) | |
| a__U71#(tt, M, N) | → | mark#(N) | a__U51#(tt, M, N) | → | mark#(N) | |
| a__U31#(tt, V1, V2) | → | a__isNat#(V1) | a__x#(N, s(M)) | → | a__and#(a__isNat(M), isNatKind(M)) | |
| a__and#(tt, X) | → | mark#(X) | a__plus#(N, 0) | → | a__U41#(a__and(a__isNat(N), isNatKind(N)), N) | |
| mark#(U41(X1, X2)) | → | a__U41#(mark(X1), X2) | a__isNat#(x(V1, V2)) | → | a__isNatKind#(V1) | |
| a__x#(N, 0) | → | a__isNat#(N) | mark#(s(X)) | → | mark#(X) | |
| a__x#(N, s(M)) | → | a__and#(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | mark#(U32(X1, X2)) | → | a__U32#(mark(X1), X2) | |
| a__isNat#(plus(V1, V2)) | → | a__U11#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) | mark#(U12(X1, X2)) | → | mark#(X1) | |
| a__U21#(tt, V1) | → | a__isNat#(V1) | a__plus#(N, s(M)) | → | a__and#(a__isNat(M), isNatKind(M)) | |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | a__isNatKind#(x(V1, V2)) | → | a__isNatKind#(V1) | |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | a__U71#(tt, M, N) | → | a__x#(mark(N), mark(M)) | |
| a__isNatKind#(plus(V1, V2)) | → | a__isNatKind#(V1) | a__U11#(tt, V1, V2) | → | a__U12#(a__isNat(V1), V2) | |
| a__isNat#(s(V1)) | → | a__isNatKind#(V1) | mark#(U31(X1, X2, X3)) | → | mark#(X1) | |
| mark#(plus(X1, X2)) | → | mark#(X1) | a__U71#(tt, M, N) | → | a__plus#(a__x(mark(N), mark(M)), mark(N)) | |
| mark#(and(X1, X2)) | → | mark#(X1) | mark#(x(X1, X2)) | → | mark#(X2) | |
| mark#(U71(X1, X2, X3)) | → | a__U71#(mark(X1), X2, X3) | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | |
| mark#(U12(X1, X2)) | → | a__U12#(mark(X1), X2) | a__plus#(N, 0) | → | a__isNat#(N) | |
| mark#(U32(X1, X2)) | → | mark#(X1) | mark#(U11(X1, X2, X3)) | → | a__U11#(mark(X1), X2, X3) | |
| mark#(U11(X1, X2, X3)) | → | mark#(X1) | a__U51#(tt, M, N) | → | a__plus#(mark(N), mark(M)) | |
| a__isNatKind#(x(V1, V2)) | → | a__and#(a__isNatKind(V1), isNatKind(V2)) | mark#(isNat(X)) | → | a__isNat#(X) | |
| mark#(x(X1, X2)) | → | a__x#(mark(X1), mark(X2)) | a__U12#(tt, V2) | → | a__isNat#(V2) | |
| mark#(U21(X1, X2)) | → | mark#(X1) | a__plus#(N, 0) | → | a__and#(a__isNat(N), isNatKind(N)) | |
| mark#(U33(X)) | → | mark#(X) | a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | |
| a__U41#(tt, N) | → | mark#(N) | a__U71#(tt, M, N) | → | mark#(M) | |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | mark#(plus(X1, X2)) | → | mark#(X2) | |
| a__isNat#(x(V1, V2)) | → | a__U31#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) | a__isNatKind#(plus(V1, V2)) | → | a__and#(a__isNatKind(V1), isNatKind(V2)) | |
| a__plus#(N, s(M)) | → | a__isNat#(M) | mark#(U41(X1, X2)) | → | mark#(X1) | |
| a__isNat#(plus(V1, V2)) | → | a__and#(a__isNatKind(V1), isNatKind(V2)) | a__plus#(N, s(M)) | → | a__and#(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | |
| mark#(U61(X)) | → | mark#(X) | mark#(U31(X1, X2, X3)) | → | a__U31#(mark(X1), X2, X3) | |
| a__x#(N, 0) | → | a__and#(a__isNat(N), isNatKind(N)) | a__x#(N, s(M)) | → | a__U71#(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | |
| a__U31#(tt, V1, V2) | → | a__U32#(a__isNat(V1), V2) | mark#(U51(X1, X2, X3)) | → | a__U51#(mark(X1), X2, X3) | |
| a__x#(N, s(M)) | → | a__isNat#(M) | mark#(U22(X)) | → | mark#(X) | |
| a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1), V1) | mark#(U21(X1, X2)) | → | a__U21#(mark(X1), X2) | |
| a__isNat#(plus(V1, V2)) | → | a__isNatKind#(V1) | a__isNat#(x(V1, V2)) | → | a__and#(a__isNatKind(V1), isNatKind(V2)) | |
| mark#(isNatKind(X)) | → | a__isNatKind#(X) | a__U51#(tt, M, N) | → | mark#(M) | |
| a__plus#(N, s(M)) | → | a__U51#(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | mark#(x(X1, X2)) | → | mark#(X1) | |
| mark#(U13(X)) | → | mark#(X) |
| a__U11(tt, V1, V2) | → | a__U12(a__isNat(V1), V2) | a__U12(tt, V2) | → | a__U13(a__isNat(V2)) | |
| a__U13(tt) | → | tt | a__U21(tt, V1) | → | a__U22(a__isNat(V1)) | |
| a__U22(tt) | → | tt | a__U31(tt, V1, V2) | → | a__U32(a__isNat(V1), V2) | |
| a__U32(tt, V2) | → | a__U33(a__isNat(V2)) | a__U33(tt) | → | tt | |
| a__U41(tt, N) | → | mark(N) | a__U51(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | |
| a__U61(tt) | → | 0 | a__U71(tt, M, N) | → | a__plus(a__x(mark(N), mark(M)), mark(N)) | |
| a__and(tt, X) | → | mark(X) | a__isNat(0) | → | tt | |
| a__isNat(plus(V1, V2)) | → | a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) | a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1), V1) | |
| a__isNat(x(V1, V2)) | → | a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) | a__isNatKind(0) | → | tt | |
| a__isNatKind(plus(V1, V2)) | → | a__and(a__isNatKind(V1), isNatKind(V2)) | a__isNatKind(s(V1)) | → | a__isNatKind(V1) | |
| a__isNatKind(x(V1, V2)) | → | a__and(a__isNatKind(V1), isNatKind(V2)) | a__plus(N, 0) | → | a__U41(a__and(a__isNat(N), isNatKind(N)), N) | |
| a__plus(N, s(M)) | → | a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | a__x(N, 0) | → | a__U61(a__and(a__isNat(N), isNatKind(N))) | |
| a__x(N, s(M)) | → | a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) | |
| mark(U12(X1, X2)) | → | a__U12(mark(X1), X2) | mark(isNat(X)) | → | a__isNat(X) | |
| mark(U13(X)) | → | a__U13(mark(X)) | mark(U21(X1, X2)) | → | a__U21(mark(X1), X2) | |
| mark(U22(X)) | → | a__U22(mark(X)) | mark(U31(X1, X2, X3)) | → | a__U31(mark(X1), X2, X3) | |
| mark(U32(X1, X2)) | → | a__U32(mark(X1), X2) | mark(U33(X)) | → | a__U33(mark(X)) | |
| mark(U41(X1, X2)) | → | a__U41(mark(X1), X2) | mark(U51(X1, X2, X3)) | → | a__U51(mark(X1), X2, X3) | |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | mark(U61(X)) | → | a__U61(mark(X)) | |
| mark(U71(X1, X2, X3)) | → | a__U71(mark(X1), X2, X3) | mark(x(X1, X2)) | → | a__x(mark(X1), mark(X2)) | |
| mark(and(X1, X2)) | → | a__and(mark(X1), X2) | mark(isNatKind(X)) | → | a__isNatKind(X) | |
| mark(tt) | → | tt | mark(s(X)) | → | s(mark(X)) | |
| mark(0) | → | 0 | a__U11(X1, X2, X3) | → | U11(X1, X2, X3) | |
| a__U12(X1, X2) | → | U12(X1, X2) | a__isNat(X) | → | isNat(X) | |
| a__U13(X) | → | U13(X) | a__U21(X1, X2) | → | U21(X1, X2) | |
| a__U22(X) | → | U22(X) | a__U31(X1, X2, X3) | → | U31(X1, X2, X3) | |
| a__U32(X1, X2) | → | U32(X1, X2) | a__U33(X) | → | U33(X) | |
| a__U41(X1, X2) | → | U41(X1, X2) | a__U51(X1, X2, X3) | → | U51(X1, X2, X3) | |
| a__plus(X1, X2) | → | plus(X1, X2) | a__U61(X) | → | U61(X) | |
| a__U71(X1, X2, X3) | → | U71(X1, X2, X3) | a__x(X1, X2) | → | x(X1, X2) | |
| a__and(X1, X2) | → | and(X1, X2) | a__isNatKind(X) | → | isNatKind(X) |
Termination of terms over the following signature is verified: a__U51, a__and, isNat, U61, U41, a__U41, a__U22, a__U71, U21, a__U21, U22, plus, isNatKind, a__plus, mark, U71, and, 0, U51, s, tt, a__isNat, a__U32, U11, a__U31, a__U13, U12, a__U12, U31, U13, a__U11, U32, a__U61, U33, a__x, a__U33, a__isNatKind, x
| a__U11#(tt, V1, V2) | → | a__isNat#(V1) | a__U32#(tt, V2) | → | a__isNat#(V2) | |
| a__U71#(tt, M, N) | → | mark#(N) | a__U51#(tt, M, N) | → | mark#(N) | |
| a__U32#(tt, V2) | → | a__U33#(a__isNat(V2)) | a__U31#(tt, V1, V2) | → | a__isNat#(V1) | |
| mark#(U13(X)) | → | a__U13#(mark(X)) | a__x#(N, s(M)) | → | a__and#(a__isNat(M), isNatKind(M)) | |
| a__and#(tt, X) | → | mark#(X) | mark#(U41(X1, X2)) | → | a__U41#(mark(X1), X2) | |
| a__plus#(N, 0) | → | a__U41#(a__and(a__isNat(N), isNatKind(N)), N) | a__isNat#(x(V1, V2)) | → | a__isNatKind#(V1) | |
| a__x#(N, 0) | → | a__isNat#(N) | mark#(s(X)) | → | mark#(X) | |
| a__x#(N, s(M)) | → | a__and#(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | a__isNat#(plus(V1, V2)) | → | a__U11#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) | |
| mark#(U32(X1, X2)) | → | a__U32#(mark(X1), X2) | mark#(U12(X1, X2)) | → | mark#(X1) | |
| a__U21#(tt, V1) | → | a__isNat#(V1) | a__plus#(N, s(M)) | → | a__and#(a__isNat(M), isNatKind(M)) | |
| mark#(U71(X1, X2, X3)) | → | mark#(X1) | mark#(U33(X)) | → | a__U33#(mark(X)) | |
| a__isNatKind#(x(V1, V2)) | → | a__isNatKind#(V1) | a__U12#(tt, V2) | → | a__U13#(a__isNat(V2)) | |
| mark#(plus(X1, X2)) | → | a__plus#(mark(X1), mark(X2)) | a__U71#(tt, M, N) | → | a__x#(mark(N), mark(M)) | |
| a__isNatKind#(plus(V1, V2)) | → | a__isNatKind#(V1) | a__isNat#(s(V1)) | → | a__isNatKind#(V1) | |
| a__U11#(tt, V1, V2) | → | a__U12#(a__isNat(V1), V2) | mark#(U31(X1, X2, X3)) | → | mark#(X1) | |
| mark#(plus(X1, X2)) | → | mark#(X1) | a__U71#(tt, M, N) | → | a__plus#(a__x(mark(N), mark(M)), mark(N)) | |
| mark#(and(X1, X2)) | → | mark#(X1) | mark#(x(X1, X2)) | → | mark#(X2) | |
| mark#(U71(X1, X2, X3)) | → | a__U71#(mark(X1), X2, X3) | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | |
| mark#(U12(X1, X2)) | → | a__U12#(mark(X1), X2) | a__plus#(N, 0) | → | a__isNat#(N) | |
| mark#(U32(X1, X2)) | → | mark#(X1) | mark#(U11(X1, X2, X3)) | → | a__U11#(mark(X1), X2, X3) | |
| a__U51#(tt, M, N) | → | a__plus#(mark(N), mark(M)) | mark#(U11(X1, X2, X3)) | → | mark#(X1) | |
| a__isNatKind#(x(V1, V2)) | → | a__and#(a__isNatKind(V1), isNatKind(V2)) | mark#(isNat(X)) | → | a__isNat#(X) | |
| mark#(x(X1, X2)) | → | a__x#(mark(X1), mark(X2)) | a__U12#(tt, V2) | → | a__isNat#(V2) | |
| mark#(U21(X1, X2)) | → | mark#(X1) | mark#(U33(X)) | → | mark#(X) | |
| a__plus#(N, 0) | → | a__and#(a__isNat(N), isNatKind(N)) | a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | |
| a__U41#(tt, N) | → | mark#(N) | a__U21#(tt, V1) | → | a__U22#(a__isNat(V1)) | |
| a__U71#(tt, M, N) | → | mark#(M) | mark#(U51(X1, X2, X3)) | → | mark#(X1) | |
| mark#(plus(X1, X2)) | → | mark#(X2) | mark#(U22(X)) | → | a__U22#(mark(X)) | |
| a__x#(N, 0) | → | a__U61#(a__and(a__isNat(N), isNatKind(N))) | a__isNat#(x(V1, V2)) | → | a__U31#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) | |
| a__plus#(N, s(M)) | → | a__isNat#(M) | a__isNatKind#(plus(V1, V2)) | → | a__and#(a__isNatKind(V1), isNatKind(V2)) | |
| a__plus#(N, s(M)) | → | a__and#(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | a__isNat#(plus(V1, V2)) | → | a__and#(a__isNatKind(V1), isNatKind(V2)) | |
| mark#(U41(X1, X2)) | → | mark#(X1) | mark#(U61(X)) | → | a__U61#(mark(X)) | |
| mark#(U61(X)) | → | mark#(X) | mark#(U31(X1, X2, X3)) | → | a__U31#(mark(X1), X2, X3) | |
| a__x#(N, 0) | → | a__and#(a__isNat(N), isNatKind(N)) | a__x#(N, s(M)) | → | a__U71#(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | |
| a__U31#(tt, V1, V2) | → | a__U32#(a__isNat(V1), V2) | mark#(U51(X1, X2, X3)) | → | a__U51#(mark(X1), X2, X3) | |
| a__x#(N, s(M)) | → | a__isNat#(M) | mark#(U22(X)) | → | mark#(X) | |
| a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1), V1) | mark#(U21(X1, X2)) | → | a__U21#(mark(X1), X2) | |
| a__isNat#(plus(V1, V2)) | → | a__isNatKind#(V1) | a__isNat#(x(V1, V2)) | → | a__and#(a__isNatKind(V1), isNatKind(V2)) | |
| mark#(isNatKind(X)) | → | a__isNatKind#(X) | a__U51#(tt, M, N) | → | mark#(M) | |
| a__plus#(N, s(M)) | → | a__U51#(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | mark#(x(X1, X2)) | → | mark#(X1) | |
| mark#(U13(X)) | → | mark#(X) |
| a__U11(tt, V1, V2) | → | a__U12(a__isNat(V1), V2) | a__U12(tt, V2) | → | a__U13(a__isNat(V2)) | |
| a__U13(tt) | → | tt | a__U21(tt, V1) | → | a__U22(a__isNat(V1)) | |
| a__U22(tt) | → | tt | a__U31(tt, V1, V2) | → | a__U32(a__isNat(V1), V2) | |
| a__U32(tt, V2) | → | a__U33(a__isNat(V2)) | a__U33(tt) | → | tt | |
| a__U41(tt, N) | → | mark(N) | a__U51(tt, M, N) | → | s(a__plus(mark(N), mark(M))) | |
| a__U61(tt) | → | 0 | a__U71(tt, M, N) | → | a__plus(a__x(mark(N), mark(M)), mark(N)) | |
| a__and(tt, X) | → | mark(X) | a__isNat(0) | → | tt | |
| a__isNat(plus(V1, V2)) | → | a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) | a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1), V1) | |
| a__isNat(x(V1, V2)) | → | a__U31(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) | a__isNatKind(0) | → | tt | |
| a__isNatKind(plus(V1, V2)) | → | a__and(a__isNatKind(V1), isNatKind(V2)) | a__isNatKind(s(V1)) | → | a__isNatKind(V1) | |
| a__isNatKind(x(V1, V2)) | → | a__and(a__isNatKind(V1), isNatKind(V2)) | a__plus(N, 0) | → | a__U41(a__and(a__isNat(N), isNatKind(N)), N) | |
| a__plus(N, s(M)) | → | a__U51(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | a__x(N, 0) | → | a__U61(a__and(a__isNat(N), isNatKind(N))) | |
| a__x(N, s(M)) | → | a__U71(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) | |
| mark(U12(X1, X2)) | → | a__U12(mark(X1), X2) | mark(isNat(X)) | → | a__isNat(X) | |
| mark(U13(X)) | → | a__U13(mark(X)) | mark(U21(X1, X2)) | → | a__U21(mark(X1), X2) | |
| mark(U22(X)) | → | a__U22(mark(X)) | mark(U31(X1, X2, X3)) | → | a__U31(mark(X1), X2, X3) | |
| mark(U32(X1, X2)) | → | a__U32(mark(X1), X2) | mark(U33(X)) | → | a__U33(mark(X)) | |
| mark(U41(X1, X2)) | → | a__U41(mark(X1), X2) | mark(U51(X1, X2, X3)) | → | a__U51(mark(X1), X2, X3) | |
| mark(plus(X1, X2)) | → | a__plus(mark(X1), mark(X2)) | mark(U61(X)) | → | a__U61(mark(X)) | |
| mark(U71(X1, X2, X3)) | → | a__U71(mark(X1), X2, X3) | mark(x(X1, X2)) | → | a__x(mark(X1), mark(X2)) | |
| mark(and(X1, X2)) | → | a__and(mark(X1), X2) | mark(isNatKind(X)) | → | a__isNatKind(X) | |
| mark(tt) | → | tt | mark(s(X)) | → | s(mark(X)) | |
| mark(0) | → | 0 | a__U11(X1, X2, X3) | → | U11(X1, X2, X3) | |
| a__U12(X1, X2) | → | U12(X1, X2) | a__isNat(X) | → | isNat(X) | |
| a__U13(X) | → | U13(X) | a__U21(X1, X2) | → | U21(X1, X2) | |
| a__U22(X) | → | U22(X) | a__U31(X1, X2, X3) | → | U31(X1, X2, X3) | |
| a__U32(X1, X2) | → | U32(X1, X2) | a__U33(X) | → | U33(X) | |
| a__U41(X1, X2) | → | U41(X1, X2) | a__U51(X1, X2, X3) | → | U51(X1, X2, X3) | |
| a__plus(X1, X2) | → | plus(X1, X2) | a__U61(X) | → | U61(X) | |
| a__U71(X1, X2, X3) | → | U71(X1, X2, X3) | a__x(X1, X2) | → | x(X1, X2) | |
| a__and(X1, X2) | → | and(X1, X2) | a__isNatKind(X) | → | isNatKind(X) |
Termination of terms over the following signature is verified: a__U51, a__and, isNat, U61, U41, a__U41, a__U22, a__U71, a__U21, U21, U22, plus, isNatKind, a__plus, mark, U71, and, 0, U51, s, tt, a__isNat, a__U32, a__U13, U11, a__U31, a__U12, U12, a__U11, U13, U31, a__U61, U32, a__x, U33, x, a__isNatKind, a__U33
| a__U32#(tt, V2) → a__isNat#(V2) | a__U11#(tt, V1, V2) → a__isNat#(V1) |
| a__U71#(tt, M, N) → mark#(N) | a__U51#(tt, M, N) → mark#(N) |
| a__U31#(tt, V1, V2) → a__isNat#(V1) | a__x#(N, s(M)) → a__and#(a__isNat(M), isNatKind(M)) |
| a__and#(tt, X) → mark#(X) | mark#(U41(X1, X2)) → a__U41#(mark(X1), X2) |
| a__plus#(N, 0) → a__U41#(a__and(a__isNat(N), isNatKind(N)), N) | a__isNat#(x(V1, V2)) → a__isNatKind#(V1) |
| mark#(s(X)) → mark#(X) | a__x#(N, 0) → a__isNat#(N) |
| a__x#(N, s(M)) → a__and#(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | a__isNat#(plus(V1, V2)) → a__U11#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) |
| mark#(U32(X1, X2)) → a__U32#(mark(X1), X2) | mark#(U12(X1, X2)) → mark#(X1) |
| a__U21#(tt, V1) → a__isNat#(V1) | mark#(U71(X1, X2, X3)) → mark#(X1) |
| a__plus#(N, s(M)) → a__and#(a__isNat(M), isNatKind(M)) | mark#(plus(X1, X2)) → a__plus#(mark(X1), mark(X2)) |
| a__isNatKind#(x(V1, V2)) → a__isNatKind#(V1) | a__U71#(tt, M, N) → a__x#(mark(N), mark(M)) |
| a__isNatKind#(plus(V1, V2)) → a__isNatKind#(V1) | a__isNat#(s(V1)) → a__isNatKind#(V1) |
| a__U11#(tt, V1, V2) → a__U12#(a__isNat(V1), V2) | mark#(U31(X1, X2, X3)) → mark#(X1) |
| a__U71#(tt, M, N) → a__plus#(a__x(mark(N), mark(M)), mark(N)) | mark#(plus(X1, X2)) → mark#(X1) |
| mark#(and(X1, X2)) → mark#(X1) | mark#(x(X1, X2)) → mark#(X2) |
| mark#(U71(X1, X2, X3)) → a__U71#(mark(X1), X2, X3) | mark#(and(X1, X2)) → a__and#(mark(X1), X2) |
| mark#(U12(X1, X2)) → a__U12#(mark(X1), X2) | a__plus#(N, 0) → a__isNat#(N) |
| mark#(U32(X1, X2)) → mark#(X1) | mark#(U11(X1, X2, X3)) → a__U11#(mark(X1), X2, X3) |
| a__U51#(tt, M, N) → a__plus#(mark(N), mark(M)) | mark#(U11(X1, X2, X3)) → mark#(X1) |
| a__isNatKind#(x(V1, V2)) → a__and#(a__isNatKind(V1), isNatKind(V2)) | mark#(isNat(X)) → a__isNat#(X) |
| mark#(x(X1, X2)) → a__x#(mark(X1), mark(X2)) | a__U12#(tt, V2) → a__isNat#(V2) |
| mark#(U21(X1, X2)) → mark#(X1) | mark#(U33(X)) → mark#(X) |
| a__plus#(N, 0) → a__and#(a__isNat(N), isNatKind(N)) | a__isNatKind#(s(V1)) → a__isNatKind#(V1) |
| a__U41#(tt, N) → mark#(N) | a__U71#(tt, M, N) → mark#(M) |
| mark#(U51(X1, X2, X3)) → mark#(X1) | mark#(plus(X1, X2)) → mark#(X2) |
| a__isNat#(x(V1, V2)) → a__U31#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) | a__plus#(N, s(M)) → a__isNat#(M) |
| a__isNatKind#(plus(V1, V2)) → a__and#(a__isNatKind(V1), isNatKind(V2)) | a__plus#(N, s(M)) → a__and#(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) |
| a__isNat#(plus(V1, V2)) → a__and#(a__isNatKind(V1), isNatKind(V2)) | mark#(U41(X1, X2)) → mark#(X1) |
| mark#(U61(X)) → mark#(X) | mark#(U31(X1, X2, X3)) → a__U31#(mark(X1), X2, X3) |
| a__x#(N, 0) → a__and#(a__isNat(N), isNatKind(N)) | a__U31#(tt, V1, V2) → a__U32#(a__isNat(V1), V2) |
| a__x#(N, s(M)) → a__U71#(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | mark#(U51(X1, X2, X3)) → a__U51#(mark(X1), X2, X3) |
| mark#(U22(X)) → mark#(X) | a__x#(N, s(M)) → a__isNat#(M) |
| a__isNat#(s(V1)) → a__U21#(a__isNatKind(V1), V1) | mark#(U21(X1, X2)) → a__U21#(mark(X1), X2) |
| a__isNat#(plus(V1, V2)) → a__isNatKind#(V1) | mark#(isNatKind(X)) → a__isNatKind#(X) |
| a__isNat#(x(V1, V2)) → a__and#(a__isNatKind(V1), isNatKind(V2)) | a__U51#(tt, M, N) → mark#(M) |
| a__plus#(N, s(M)) → a__U51#(a__and(a__and(a__isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N) | mark#(x(X1, X2)) → mark#(X1) |
| mark#(U13(X)) → mark#(X) |