The TRS could not be proven terminating. The proof attempt took 60024 ms.
Problem 1 remains open; application of the following processors failed [DependencyGraph (60002ms)].
| mark#(U32(X)) | → | U32#(mark(X)) | active#(U52(tt, V2)) | → | isNatList#(V2) | |
| U11#(mark(X1), X2) | → | U11#(X1, X2) | U52#(X1, active(X2)) | → | U52#(X1, X2) | |
| active#(isNatIListKind(cons(V1, V2))) | → | isNatKind#(V1) | U21#(X1, active(X2)) | → | U21#(X1, X2) | |
| isNat#(active(X)) | → | isNat#(X) | mark#(s(X)) | → | mark#(X) | |
| mark#(U11(X1, X2)) | → | active#(U11(mark(X1), X2)) | length#(mark(X)) | → | length#(X) | |
| mark#(U53(X)) | → | active#(U53(mark(X))) | and#(mark(X1), X2) | → | and#(X1, X2) | |
| active#(U21(tt, V1)) | → | isNat#(V1) | mark#(U12(X)) | → | active#(U12(mark(X))) | |
| U42#(mark(X1), X2) | → | U42#(X1, X2) | U11#(X1, mark(X2)) | → | U11#(X1, X2) | |
| active#(U12(tt)) | → | mark#(tt) | U41#(X1, active(X2), X3) | → | U41#(X1, X2, X3) | |
| mark#(U41(X1, X2, X3)) | → | mark#(X1) | U53#(mark(X)) | → | U53#(X) | |
| mark#(U22(X)) | → | U22#(mark(X)) | mark#(U22(X)) | → | active#(U22(mark(X))) | |
| mark#(U53(X)) | → | U53#(mark(X)) | mark#(U32(X)) | → | mark#(X) | |
| U32#(mark(X)) | → | U32#(X) | mark#(U21(X1, X2)) | → | active#(U21(mark(X1), X2)) | |
| active#(isNatIListKind(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) | mark#(U42(X1, X2)) | → | U42#(mark(X1), X2) | |
| cons#(X1, mark(X2)) | → | cons#(X1, X2) | U41#(X1, X2, active(X3)) | → | U41#(X1, X2, X3) | |
| isNatKind#(mark(X)) | → | isNatKind#(X) | active#(U41(tt, V1, V2)) | → | U42#(isNat(V1), V2) | |
| mark#(isNatKind(X)) | → | isNatKind#(X) | U52#(X1, mark(X2)) | → | U52#(X1, X2) | |
| active#(isNatIList(cons(V1, V2))) | → | isNatIListKind#(V2) | mark#(U42(X1, X2)) | → | mark#(X1) | |
| active#(length(cons(N, L))) | → | and#(isNatList(L), isNatIListKind(L)) | U21#(active(X1), X2) | → | U21#(X1, X2) | |
| cons#(active(X1), X2) | → | cons#(X1, X2) | mark#(isNatList(X)) | → | isNatList#(X) | |
| active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | isNatList#(mark(X)) | → | isNatList#(X) | |
| mark#(U41(X1, X2, X3)) | → | active#(U41(mark(X1), X2, X3)) | U42#(active(X1), X2) | → | U42#(X1, X2) | |
| active#(U11(tt, V1)) | → | isNatList#(V1) | U52#(mark(X1), X2) | → | U52#(X1, X2) | |
| active#(U42(tt, V2)) | → | isNatIList#(V2) | U61#(active(X1), X2) | → | U61#(X1, X2) | |
| mark#(U52(X1, X2)) | → | U52#(mark(X1), X2) | mark#(s(X)) | → | s#(mark(X)) | |
| U12#(mark(X)) | → | U12#(X) | active#(U41(tt, V1, V2)) | → | mark#(U42(isNat(V1), V2)) | |
| active#(U11(tt, V1)) | → | U12#(isNatList(V1)) | U61#(mark(X1), X2) | → | U61#(X1, X2) | |
| U22#(mark(X)) | → | U22#(X) | active#(U61(tt, L)) | → | s#(length(L)) | |
| isNatList#(active(X)) | → | isNatList#(X) | U51#(X1, mark(X2), X3) | → | U51#(X1, X2, X3) | |
| active#(isNat(0)) | → | mark#(tt) | mark#(and(X1, X2)) | → | and#(mark(X1), X2) | |
| active#(length(cons(N, L))) | → | and#(isNat(N), isNatKind(N)) | active#(isNatIListKind(cons(V1, V2))) | → | isNatIListKind#(V2) | |
| active#(isNatIList(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) | mark#(isNatIList(X)) | → | isNatIList#(X) | |
| mark#(U61(X1, X2)) | → | active#(U61(mark(X1), X2)) | isNatIList#(mark(X)) | → | isNatIList#(X) | |
| active#(U21(tt, V1)) | → | U22#(isNat(V1)) | cons#(X1, active(X2)) | → | cons#(X1, X2) | |
| isNatKind#(active(X)) | → | isNatKind#(X) | U12#(active(X)) | → | U12#(X) | |
| active#(isNat(length(V1))) | → | U11#(isNatIListKind(V1), V1) | U22#(active(X)) | → | U22#(X) | |
| U51#(active(X1), X2, X3) | → | U51#(X1, X2, X3) | U43#(active(X)) | → | U43#(X) | |
| mark#(U61(X1, X2)) | → | mark#(X1) | active#(isNatKind(length(V1))) | → | mark#(isNatIListKind(V1)) | |
| mark#(U41(X1, X2, X3)) | → | U41#(mark(X1), X2, X3) | and#(X1, active(X2)) | → | and#(X1, X2) | |
| active#(length(cons(N, L))) | → | isNatKind#(N) | mark#(U21(X1, X2)) | → | mark#(X1) | |
| mark#(U21(X1, X2)) | → | U21#(mark(X1), X2) | U61#(X1, mark(X2)) | → | U61#(X1, X2) | |
| mark#(isNat(X)) | → | isNat#(X) | isNat#(mark(X)) | → | isNat#(X) | |
| active#(length(cons(N, L))) | → | and#(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))) | active#(length(cons(N, L))) | → | isNatList#(L) | |
| active#(isNatList(nil)) | → | mark#(tt) | mark#(nil) | → | active#(nil) | |
| active#(length(cons(N, L))) | → | isNat#(N) | mark#(isNatIListKind(X)) | → | active#(isNatIListKind(X)) | |
| active#(isNatIList(zeros)) | → | mark#(tt) | mark#(cons(X1, X2)) | → | cons#(mark(X1), X2) | |
| and#(X1, mark(X2)) | → | and#(X1, X2) | mark#(length(X)) | → | length#(mark(X)) | |
| U43#(mark(X)) | → | U43#(X) | active#(isNat(s(V1))) | → | mark#(U21(isNatKind(V1), V1)) | |
| s#(mark(X)) | → | s#(X) | active#(length(cons(N, L))) | → | mark#(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | |
| mark#(U31(X1, X2)) | → | U31#(mark(X1), X2) | mark#(U51(X1, X2, X3)) | → | U51#(mark(X1), X2, X3) | |
| mark#(U32(X)) | → | active#(U32(mark(X))) | active#(isNatIList(V)) | → | U31#(isNatIListKind(V), V) | |
| mark#(U42(X1, X2)) | → | active#(U42(mark(X1), X2)) | mark#(length(X)) | → | active#(length(mark(X))) | |
| mark#(U43(X)) | → | U43#(mark(X)) | active#(isNatKind(0)) | → | mark#(tt) | |
| mark#(U11(X1, X2)) | → | mark#(X1) | isNatIListKind#(active(X)) | → | isNatIListKind#(X) | |
| active#(isNatIList(V)) | → | isNatIListKind#(V) | active#(length(nil)) | → | mark#(0) | |
| active#(U11(tt, V1)) | → | mark#(U12(isNatList(V1))) | active#(U22(tt)) | → | mark#(tt) | |
| active#(U61(tt, L)) | → | mark#(s(length(L))) | active#(isNat(length(V1))) | → | mark#(U11(isNatIListKind(V1), V1)) | |
| mark#(U53(X)) | → | mark#(X) | active#(U53(tt)) | → | mark#(tt) | |
| U11#(active(X1), X2) | → | U11#(X1, X2) | U51#(X1, X2, active(X3)) | → | U51#(X1, X2, X3) | |
| active#(U51(tt, V1, V2)) | → | U52#(isNat(V1), V2) | mark#(cons(X1, X2)) | → | mark#(X1) | |
| active#(U52(tt, V2)) | → | U53#(isNatList(V2)) | active#(U61(tt, L)) | → | length#(L) | |
| active#(isNatKind(length(V1))) | → | isNatIListKind#(V1) | active#(isNat(s(V1))) | → | isNatKind#(V1) | |
| mark#(and(X1, X2)) | → | mark#(X1) | U11#(X1, active(X2)) | → | U11#(X1, X2) | |
| active#(isNatList(cons(V1, V2))) | → | isNatIListKind#(V2) | active#(U31(tt, V)) | → | U32#(isNatList(V)) | |
| active#(isNatList(cons(V1, V2))) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | mark#(tt) | → | active#(tt) | |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | U31#(X1, active(X2)) | → | U31#(X1, X2) | |
| mark#(U61(X1, X2)) | → | U61#(mark(X1), X2) | active#(isNatList(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) | |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | U21#(X1, mark(X2)) | → | U21#(X1, X2) | |
| U61#(X1, active(X2)) | → | U61#(X1, X2) | active#(length(cons(N, L))) | → | U61#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | |
| mark#(length(X)) | → | mark#(X) | mark#(zeros) | → | active#(zeros) | |
| active#(isNatKind(s(V1))) | → | mark#(isNatKind(V1)) | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | |
| active#(isNat(length(V1))) | → | isNatIListKind#(V1) | mark#(0) | → | active#(0) | |
| U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | mark#(s(X)) | → | active#(s(mark(X))) | |
| U41#(X1, X2, mark(X3)) | → | U41#(X1, X2, X3) | U52#(active(X1), X2) | → | U52#(X1, X2) | |
| mark#(U43(X)) | → | active#(U43(mark(X))) | mark#(U22(X)) | → | mark#(X) | |
| mark#(U43(X)) | → | mark#(X) | active#(isNatIListKind(nil)) | → | mark#(tt) | |
| active#(zeros) | → | cons#(0, zeros) | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | |
| isNatIListKind#(mark(X)) | → | isNatIListKind#(X) | mark#(isNatIListKind(X)) | → | isNatIListKind#(X) | |
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | active#(U21(tt, V1)) | → | mark#(U22(isNat(V1))) | |
| active#(isNatIListKind(zeros)) | → | mark#(tt) | mark#(isNatKind(X)) | → | active#(isNatKind(X)) | |
| active#(U51(tt, V1, V2)) | → | isNat#(V1) | active#(isNatIList(V)) | → | mark#(U31(isNatIListKind(V), V)) | |
| U53#(active(X)) | → | U53#(X) | active#(U42(tt, V2)) | → | mark#(U43(isNatIList(V2))) | |
| U51#(X1, active(X2), X3) | → | U51#(X1, X2, X3) | active#(U42(tt, V2)) | → | U43#(isNatIList(V2)) | |
| length#(active(X)) | → | length#(X) | U31#(active(X1), X2) | → | U31#(X1, X2) | |
| U42#(X1, mark(X2)) | → | U42#(X1, X2) | active#(U31(tt, V)) | → | mark#(U32(isNatList(V))) | |
| mark#(U31(X1, X2)) | → | mark#(X1) | active#(U52(tt, V2)) | → | mark#(U53(isNatList(V2))) | |
| active#(U32(tt)) | → | mark#(tt) | mark#(isNatIList(X)) | → | active#(isNatIList(X)) | |
| mark#(U12(X)) | → | U12#(mark(X)) | active#(isNatList(cons(V1, V2))) | → | mark#(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | |
| isNatIList#(active(X)) | → | isNatIList#(X) | active#(U31(tt, V)) | → | isNatList#(V) | |
| active#(length(cons(N, L))) | → | isNatIListKind#(L) | U41#(active(X1), X2, X3) | → | U41#(X1, X2, X3) | |
| active#(U43(tt)) | → | mark#(tt) | active#(isNatIList(cons(V1, V2))) | → | mark#(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | |
| and#(active(X1), X2) | → | and#(X1, X2) | mark#(U52(X1, X2)) | → | active#(U52(mark(X1), X2)) | |
| mark#(isNat(X)) | → | active#(isNat(X)) | U42#(X1, active(X2)) | → | U42#(X1, X2) | |
| U31#(X1, mark(X2)) | → | U31#(X1, X2) | active#(U41(tt, V1, V2)) | → | isNat#(V1) | |
| active#(isNatIListKind(cons(V1, V2))) | → | mark#(and(isNatKind(V1), isNatIListKind(V2))) | mark#(U11(X1, X2)) | → | U11#(mark(X1), X2) | |
| mark#(and(X1, X2)) | → | active#(and(mark(X1), X2)) | U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | |
| active#(U51(tt, V1, V2)) | → | mark#(U52(isNat(V1), V2)) | active#(isNatKind(s(V1))) | → | isNatKind#(V1) | |
| U32#(active(X)) | → | U32#(X) | mark#(U12(X)) | → | mark#(X) | |
| mark#(U52(X1, X2)) | → | mark#(X1) | active#(isNatIList(cons(V1, V2))) | → | isNatKind#(V1) | |
| U51#(X1, X2, mark(X3)) | → | U51#(X1, X2, X3) | active#(and(tt, X)) | → | mark#(X) | |
| U41#(X1, mark(X2), X3) | → | U41#(X1, X2, X3) | U31#(mark(X1), X2) | → | U31#(X1, X2) | |
| active#(isNatIList(cons(V1, V2))) | → | U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | U21#(mark(X1), X2) | → | U21#(X1, X2) | |
| active#(isNatList(cons(V1, V2))) | → | isNatKind#(V1) | active#(zeros) | → | mark#(cons(0, zeros)) | |
| s#(active(X)) | → | s#(X) | mark#(isNatList(X)) | → | active#(isNatList(X)) |
| active(zeros) | → | mark(cons(0, zeros)) | active(U11(tt, V1)) | → | mark(U12(isNatList(V1))) | |
| active(U12(tt)) | → | mark(tt) | active(U21(tt, V1)) | → | mark(U22(isNat(V1))) | |
| active(U22(tt)) | → | mark(tt) | active(U31(tt, V)) | → | mark(U32(isNatList(V))) | |
| active(U32(tt)) | → | mark(tt) | active(U41(tt, V1, V2)) | → | mark(U42(isNat(V1), V2)) | |
| active(U42(tt, V2)) | → | mark(U43(isNatIList(V2))) | active(U43(tt)) | → | mark(tt) | |
| active(U51(tt, V1, V2)) | → | mark(U52(isNat(V1), V2)) | active(U52(tt, V2)) | → | mark(U53(isNatList(V2))) | |
| active(U53(tt)) | → | mark(tt) | active(U61(tt, L)) | → | mark(s(length(L))) | |
| active(and(tt, X)) | → | mark(X) | active(isNat(0)) | → | mark(tt) | |
| active(isNat(length(V1))) | → | mark(U11(isNatIListKind(V1), V1)) | active(isNat(s(V1))) | → | mark(U21(isNatKind(V1), V1)) | |
| active(isNatIList(V)) | → | mark(U31(isNatIListKind(V), V)) | active(isNatIList(zeros)) | → | mark(tt) | |
| active(isNatIList(cons(V1, V2))) | → | mark(U41(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | active(isNatIListKind(nil)) | → | mark(tt) | |
| active(isNatIListKind(zeros)) | → | mark(tt) | active(isNatIListKind(cons(V1, V2))) | → | mark(and(isNatKind(V1), isNatIListKind(V2))) | |
| active(isNatKind(0)) | → | mark(tt) | active(isNatKind(length(V1))) | → | mark(isNatIListKind(V1)) | |
| active(isNatKind(s(V1))) | → | mark(isNatKind(V1)) | active(isNatList(nil)) | → | mark(tt) | |
| active(isNatList(cons(V1, V2))) | → | mark(U51(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | active(length(nil)) | → | mark(0) | |
| active(length(cons(N, L))) | → | mark(U61(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | mark(zeros) | → | active(zeros) | |
| mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | mark(0) | → | active(0) | |
| mark(U11(X1, X2)) | → | active(U11(mark(X1), X2)) | mark(tt) | → | active(tt) | |
| mark(U12(X)) | → | active(U12(mark(X))) | mark(isNatList(X)) | → | active(isNatList(X)) | |
| mark(U21(X1, X2)) | → | active(U21(mark(X1), X2)) | mark(U22(X)) | → | active(U22(mark(X))) | |
| mark(isNat(X)) | → | active(isNat(X)) | mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | |
| mark(U32(X)) | → | active(U32(mark(X))) | mark(U41(X1, X2, X3)) | → | active(U41(mark(X1), X2, X3)) | |
| mark(U42(X1, X2)) | → | active(U42(mark(X1), X2)) | mark(U43(X)) | → | active(U43(mark(X))) | |
| mark(isNatIList(X)) | → | active(isNatIList(X)) | mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) | |
| mark(U52(X1, X2)) | → | active(U52(mark(X1), X2)) | mark(U53(X)) | → | active(U53(mark(X))) | |
| mark(U61(X1, X2)) | → | active(U61(mark(X1), X2)) | mark(s(X)) | → | active(s(mark(X))) | |
| mark(length(X)) | → | active(length(mark(X))) | mark(and(X1, X2)) | → | active(and(mark(X1), X2)) | |
| mark(isNatIListKind(X)) | → | active(isNatIListKind(X)) | mark(isNatKind(X)) | → | active(isNatKind(X)) | |
| mark(nil) | → | active(nil) | cons(mark(X1), X2) | → | cons(X1, X2) | |
| cons(X1, mark(X2)) | → | cons(X1, X2) | cons(active(X1), X2) | → | cons(X1, X2) | |
| cons(X1, active(X2)) | → | cons(X1, X2) | U11(mark(X1), X2) | → | U11(X1, X2) | |
| U11(X1, mark(X2)) | → | U11(X1, X2) | U11(active(X1), X2) | → | U11(X1, X2) | |
| U11(X1, active(X2)) | → | U11(X1, X2) | U12(mark(X)) | → | U12(X) | |
| U12(active(X)) | → | U12(X) | isNatList(mark(X)) | → | isNatList(X) | |
| isNatList(active(X)) | → | isNatList(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) | isNat(mark(X)) | → | isNat(X) | |
| isNat(active(X)) | → | isNat(X) | U31(mark(X1), X2) | → | U31(X1, X2) | |
| U31(X1, mark(X2)) | → | U31(X1, X2) | U31(active(X1), X2) | → | U31(X1, X2) | |
| U31(X1, active(X2)) | → | U31(X1, X2) | U32(mark(X)) | → | U32(X) | |
| U32(active(X)) | → | U32(X) | U41(mark(X1), X2, X3) | → | U41(X1, X2, X3) | |
| U41(X1, mark(X2), X3) | → | U41(X1, X2, X3) | U41(X1, X2, mark(X3)) | → | U41(X1, X2, X3) | |
| U41(active(X1), X2, X3) | → | U41(X1, X2, X3) | U41(X1, active(X2), X3) | → | U41(X1, X2, X3) | |
| U41(X1, X2, active(X3)) | → | U41(X1, X2, X3) | U42(mark(X1), X2) | → | U42(X1, X2) | |
| U42(X1, mark(X2)) | → | U42(X1, X2) | U42(active(X1), X2) | → | U42(X1, X2) | |
| U42(X1, active(X2)) | → | U42(X1, X2) | U43(mark(X)) | → | U43(X) | |
| U43(active(X)) | → | U43(X) | isNatIList(mark(X)) | → | isNatIList(X) | |
| isNatIList(active(X)) | → | isNatIList(X) | 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) | U52(mark(X1), X2) | → | U52(X1, X2) | |
| U52(X1, mark(X2)) | → | U52(X1, X2) | U52(active(X1), X2) | → | U52(X1, X2) | |
| U52(X1, active(X2)) | → | U52(X1, X2) | U53(mark(X)) | → | U53(X) | |
| U53(active(X)) | → | U53(X) | U61(mark(X1), X2) | → | U61(X1, X2) | |
| U61(X1, mark(X2)) | → | U61(X1, X2) | U61(active(X1), X2) | → | U61(X1, X2) | |
| U61(X1, active(X2)) | → | U61(X1, X2) | s(mark(X)) | → | s(X) | |
| s(active(X)) | → | s(X) | length(mark(X)) | → | length(X) | |
| length(active(X)) | → | length(X) | 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) | isNatIListKind(mark(X)) | → | isNatIListKind(X) | |
| isNatIListKind(active(X)) | → | isNatIListKind(X) | isNatKind(mark(X)) | → | isNatKind(X) | |
| isNatKind(active(X)) | → | isNatKind(X) |
Termination of terms over the following signature is verified: isNatIListKind, isNat, U43, U61, U42, U41, length, U21, U22, cons, isNatIList, isNatKind, mark, and, 0, isNatList, U51, s, tt, zeros, U53, U52, U11, active, U12, U31, U32, nil