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#(cons(X1, X2)) | → | proper#(X1) | U85#(ok(X1), ok(X2)) | → | U85#(X1, X2) | |
| proper#(length(X)) | → | length#(proper(X)) | active#(U51(X1, X2)) | → | active#(X1) | |
| active#(U94(X1, X2)) | → | active#(X1) | active#(U93(X1, X2, X3)) | → | active#(X1) | |
| U11#(mark(X1), X2) | → | U11#(X1, X2) | active#(isNatIListKind(cons(V1, V2))) | → | U51#(isNatKind(V1), V2) | |
| proper#(U92(X1, X2, X3)) | → | proper#(X1) | active#(isNatList(cons(V1, V2))) | → | U81#(isNatKind(V1), V1, V2) | |
| U51#(mark(X1), X2) | → | U51#(X1, X2) | U91#(mark(X1), X2, X3) | → | U91#(X1, X2, X3) | |
| active#(isNatIListKind(cons(V1, V2))) | → | isNatKind#(V1) | active#(isNatIList(cons(V1, V2))) | → | U41#(isNatKind(V1), V1, V2) | |
| top#(mark(X)) | → | proper#(X) | length#(mark(X)) | → | length#(X) | |
| isNatKind#(ok(X)) | → | isNatKind#(X) | U43#(mark(X1), X2, X3) | → | U43#(X1, X2, X3) | |
| U86#(mark(X)) | → | U86#(X) | active#(U81(X1, X2, X3)) | → | active#(X1) | |
| active#(U41(tt, V1, V2)) | → | isNatKind#(V1) | active#(U82(tt, V1, V2)) | → | U83#(isNatIListKind(V2), V1, V2) | |
| active#(length(X)) | → | active#(X) | proper#(U43(X1, X2, X3)) | → | proper#(X3) | |
| active#(U21(tt, V1)) | → | isNatKind#(V1) | proper#(U42(X1, X2, X3)) | → | U42#(proper(X1), proper(X2), proper(X3)) | |
| active#(U86(X)) | → | U86#(active(X)) | active#(U85(tt, V2)) | → | isNatList#(V2) | |
| proper#(U23(X)) | → | U23#(proper(X)) | U43#(ok(X1), ok(X2), ok(X3)) | → | U43#(X1, X2, X3) | |
| proper#(U82(X1, X2, X3)) | → | proper#(X1) | active#(U32(tt, V)) | → | U33#(isNatList(V)) | |
| proper#(U43(X1, X2, X3)) | → | proper#(X1) | proper#(isNatKind(X)) | → | isNatKind#(proper(X)) | |
| active#(U33(X)) | → | active#(X) | active#(U43(tt, V1, V2)) | → | U44#(isNatIListKind(V2), V1, V2) | |
| U45#(ok(X1), ok(X2)) | → | U45#(X1, X2) | proper#(U44(X1, X2, X3)) | → | U44#(proper(X1), proper(X2), proper(X3)) | |
| active#(U41(X1, X2, X3)) | → | U41#(active(X1), X2, X3) | active#(U44(tt, V1, V2)) | → | U45#(isNat(V1), V2) | |
| proper#(U45(X1, X2)) | → | U45#(proper(X1), proper(X2)) | active#(U91(tt, L, N)) | → | isNatIListKind#(L) | |
| active#(U84(tt, V1, V2)) | → | U85#(isNat(V1), V2) | active#(U45(tt, V2)) | → | U46#(isNatIList(V2)) | |
| active#(U44(X1, X2, X3)) | → | active#(X1) | proper#(U41(X1, X2, X3)) | → | proper#(X1) | |
| U93#(ok(X1), ok(X2), ok(X3)) | → | U93#(X1, X2, X3) | active#(U52(X)) | → | active#(X) | |
| active#(U31(X1, X2)) | → | active#(X1) | active#(U22(X1, X2)) | → | active#(X1) | |
| active#(isNatKind(s(V1))) | → | U71#(isNatKind(V1)) | active#(U81(tt, V1, V2)) | → | U82#(isNatKind(V1), V1, V2) | |
| proper#(isNatList(X)) | → | proper#(X) | active#(U13(X)) | → | active#(X) | |
| proper#(U51(X1, X2)) | → | proper#(X2) | active#(U93(tt, L, N)) | → | isNatKind#(N) | |
| proper#(U91(X1, X2, X3)) | → | proper#(X3) | proper#(U71(X)) | → | U71#(proper(X)) | |
| active#(U83(X1, X2, X3)) | → | U83#(active(X1), X2, X3) | proper#(U22(X1, X2)) | → | U22#(proper(X1), proper(X2)) | |
| U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) | active#(U21(X1, X2)) | → | U21#(active(X1), X2) | |
| active#(U52(X)) | → | U52#(active(X)) | active#(U12(tt, V1)) | → | isNatList#(V1) | |
| proper#(isNat(X)) | → | proper#(X) | proper#(U91(X1, X2, X3)) | → | U91#(proper(X1), proper(X2), proper(X3)) | |
| active#(U94(tt, L)) | → | s#(length(L)) | U33#(mark(X)) | → | U33#(X) | |
| active#(U11(X1, X2)) | → | U11#(active(X1), X2) | proper#(U92(X1, X2, X3)) | → | proper#(X3) | |
| active#(U83(X1, X2, X3)) | → | active#(X1) | active#(U92(tt, L, N)) | → | isNat#(N) | |
| U22#(ok(X1), ok(X2)) | → | U22#(X1, X2) | active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | |
| U91#(ok(X1), ok(X2), ok(X3)) | → | U91#(X1, X2, X3) | active#(U51(tt, V2)) | → | isNatIListKind#(V2) | |
| proper#(U31(X1, X2)) | → | U31#(proper(X1), proper(X2)) | proper#(U21(X1, X2)) | → | proper#(X1) | |
| active#(U41(tt, V1, V2)) | → | U42#(isNatKind(V1), V1, V2) | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) | |
| active#(U31(tt, V)) | → | U32#(isNatIListKind(V), V) | proper#(U81(X1, X2, X3)) | → | U81#(proper(X1), proper(X2), proper(X3)) | |
| active#(U22(X1, X2)) | → | U22#(active(X1), X2) | U12#(ok(X1), ok(X2)) | → | U12#(X1, X2) | |
| isNatIListKind#(ok(X)) | → | isNatIListKind#(X) | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) | |
| proper#(U12(X1, X2)) | → | proper#(X2) | active#(U42(X1, X2, X3)) | → | active#(X1) | |
| proper#(U11(X1, X2)) | → | U11#(proper(X1), proper(X2)) | active#(U21(tt, V1)) | → | U22#(isNatKind(V1), V1) | |
| proper#(U51(X1, X2)) | → | U51#(proper(X1), proper(X2)) | U85#(mark(X1), X2) | → | U85#(X1, X2) | |
| proper#(U94(X1, X2)) | → | proper#(X2) | active#(U51(X1, X2)) | → | U51#(active(X1), X2) | |
| length#(ok(X)) | → | length#(X) | U12#(mark(X1), X2) | → | U12#(X1, X2) | |
| proper#(U81(X1, X2, X3)) | → | proper#(X2) | active#(U85(tt, V2)) | → | U86#(isNatList(V2)) | |
| proper#(U32(X1, X2)) | → | proper#(X1) | proper#(U41(X1, X2, X3)) | → | U41#(proper(X1), proper(X2), proper(X3)) | |
| active#(U85(X1, X2)) | → | U85#(active(X1), X2) | U71#(mark(X)) | → | U71#(X) | |
| active#(U11(tt, V1)) | → | isNatIListKind#(V1) | active#(U61(X)) | → | U61#(active(X)) | |
| active#(U92(X1, X2, X3)) | → | active#(X1) | U52#(ok(X)) | → | U52#(X) | |
| active#(U22(tt, V1)) | → | U23#(isNat(V1)) | active#(U93(X1, X2, X3)) | → | U93#(active(X1), X2, X3) | |
| proper#(s(X)) | → | proper#(X) | U32#(ok(X1), ok(X2)) | → | U32#(X1, X2) | |
| active#(U51(tt, V2)) | → | U52#(isNatIListKind(V2)) | active#(U13(X)) | → | U13#(active(X)) | |
| proper#(isNatList(X)) | → | isNatList#(proper(X)) | proper#(U45(X1, X2)) | → | proper#(X2) | |
| active#(isNat(length(V1))) | → | U11#(isNatIListKind(V1), V1) | active#(U33(X)) | → | U33#(active(X)) | |
| proper#(U85(X1, X2)) | → | proper#(X2) | proper#(U32(X1, X2)) | → | U32#(proper(X1), proper(X2)) | |
| U84#(ok(X1), ok(X2), ok(X3)) | → | U84#(X1, X2, X3) | U33#(ok(X)) | → | U33#(X) | |
| active#(cons(X1, X2)) | → | active#(X1) | proper#(U81(X1, X2, X3)) | → | proper#(X3) | |
| active#(U43(tt, V1, V2)) | → | isNatIListKind#(V2) | active#(U82(X1, X2, X3)) | → | active#(X1) | |
| U44#(ok(X1), ok(X2), ok(X3)) | → | U44#(X1, X2, X3) | active#(U86(X)) | → | active#(X) | |
| proper#(U31(X1, X2)) | → | proper#(X2) | proper#(U92(X1, X2, X3)) | → | U92#(proper(X1), proper(X2), proper(X3)) | |
| proper#(U61(X)) | → | U61#(proper(X)) | proper#(U51(X1, X2)) | → | proper#(X1) | |
| active#(length(cons(N, L))) | → | isNatList#(L) | proper#(U43(X1, X2, X3)) | → | proper#(X2) | |
| active#(U83(tt, V1, V2)) | → | U84#(isNatIListKind(V2), V1, V2) | proper#(U85(X1, X2)) | → | U85#(proper(X1), proper(X2)) | |
| active#(U11(tt, V1)) | → | U12#(isNatIListKind(V1), V1) | proper#(U81(X1, X2, X3)) | → | proper#(X1) | |
| active#(U44(X1, X2, X3)) | → | U44#(active(X1), X2, X3) | proper#(U42(X1, X2, X3)) | → | proper#(X2) | |
| active#(length(cons(N, L))) | → | U91#(isNatList(L), L, N) | U94#(mark(X1), X2) | → | U94#(X1, X2) | |
| proper#(U12(X1, X2)) | → | proper#(X1) | proper#(U61(X)) | → | proper#(X) | |
| active#(isNatKind(length(V1))) | → | U61#(isNatIListKind(V1)) | proper#(U84(X1, X2, X3)) | → | proper#(X3) | |
| proper#(U84(X1, X2, X3)) | → | proper#(X1) | proper#(U33(X)) | → | U33#(proper(X)) | |
| s#(mark(X)) | → | s#(X) | proper#(U84(X1, X2, X3)) | → | proper#(X2) | |
| active#(U31(tt, V)) | → | isNatIListKind#(V) | proper#(U86(X)) | → | proper#(X) | |
| active#(U45(X1, X2)) | → | U45#(active(X1), X2) | active#(s(X)) | → | active#(X) | |
| proper#(U46(X)) | → | proper#(X) | U83#(mark(X1), X2, X3) | → | U83#(X1, X2, X3) | |
| U52#(mark(X)) | → | U52#(X) | U92#(ok(X1), ok(X2), ok(X3)) | → | U92#(X1, X2, X3) | |
| active#(U61(X)) | → | active#(X) | active#(isNatIList(V)) | → | U31#(isNatIListKind(V), V) | |
| U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) | active#(U82(X1, X2, X3)) | → | U82#(active(X1), X2, X3) | |
| U42#(ok(X1), ok(X2), ok(X3)) | → | U42#(X1, X2, X3) | proper#(U44(X1, X2, X3)) | → | proper#(X2) | |
| proper#(U33(X)) | → | proper#(X) | proper#(U11(X1, X2)) | → | proper#(X1) | |
| proper#(U71(X)) | → | proper#(X) | active#(isNatIList(V)) | → | isNatIListKind#(V) | |
| proper#(U91(X1, X2, X3)) | → | proper#(X2) | proper#(U52(X)) | → | U52#(proper(X)) | |
| active#(U42(X1, X2, X3)) | → | U42#(active(X1), X2, X3) | active#(U32(tt, V)) | → | isNatList#(V) | |
| proper#(U91(X1, X2, X3)) | → | proper#(X1) | proper#(U41(X1, X2, X3)) | → | proper#(X2) | |
| proper#(U32(X1, X2)) | → | proper#(X2) | proper#(isNatKind(X)) | → | proper#(X) | |
| proper#(U94(X1, X2)) | → | proper#(X1) | proper#(U11(X1, X2)) | → | proper#(X2) | |
| active#(U42(tt, V1, V2)) | → | isNatIListKind#(V2) | active#(U83(tt, V1, V2)) | → | isNatIListKind#(V2) | |
| proper#(U23(X)) | → | proper#(X) | active#(U22(tt, V1)) | → | isNat#(V1) | |
| U44#(mark(X1), X2, X3) | → | U44#(X1, X2, X3) | active#(U42(tt, V1, V2)) | → | U43#(isNatIListKind(V2), V1, V2) | |
| active#(U46(X)) | → | active#(X) | isNat#(ok(X)) | → | isNat#(X) | |
| proper#(U46(X)) | → | U46#(proper(X)) | proper#(U94(X1, X2)) | → | U94#(proper(X1), proper(X2)) | |
| isNatIList#(ok(X)) | → | isNatIList#(X) | active#(U41(X1, X2, X3)) | → | active#(X1) | |
| active#(U12(X1, X2)) | → | U12#(active(X1), X2) | active#(isNatKind(length(V1))) | → | isNatIListKind#(V1) | |
| active#(isNat(s(V1))) | → | isNatKind#(V1) | proper#(U13(X)) | → | U13#(proper(X)) | |
| proper#(U42(X1, X2, X3)) | → | proper#(X3) | active#(U91(tt, L, N)) | → | U92#(isNatIListKind(L), L, N) | |
| proper#(U93(X1, X2, X3)) | → | proper#(X1) | active#(U93(tt, L, N)) | → | U94#(isNatKind(N), L) | |
| proper#(isNatIListKind(X)) | → | proper#(X) | U22#(mark(X1), X2) | → | U22#(X1, X2) | |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | proper#(U44(X1, X2, X3)) | → | proper#(X1) | |
| active#(U43(X1, X2, X3)) | → | active#(X1) | top#(ok(X)) | → | active#(X) | |
| active#(U91(X1, X2, X3)) | → | U91#(active(X1), X2, X3) | U82#(mark(X1), X2, X3) | → | U82#(X1, X2, X3) | |
| proper#(U93(X1, X2, X3)) | → | proper#(X3) | proper#(U93(X1, X2, X3)) | → | U93#(proper(X1), proper(X2), proper(X3)) | |
| U45#(mark(X1), X2) | → | U45#(X1, X2) | U83#(ok(X1), ok(X2), ok(X3)) | → | U83#(X1, X2, X3) | |
| U94#(ok(X1), ok(X2)) | → | U94#(X1, X2) | U13#(mark(X)) | → | U13#(X) | |
| active#(U94(tt, L)) | → | length#(L) | proper#(isNat(X)) | → | isNat#(proper(X)) | |
| active#(isNat(length(V1))) | → | isNatIListKind#(V1) | U51#(ok(X1), ok(X2)) | → | U51#(X1, X2) | |
| proper#(U45(X1, X2)) | → | proper#(X1) | U61#(mark(X)) | → | U61#(X) | |
| active#(U12(X1, X2)) | → | active#(X1) | proper#(U42(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#(length(X)) | → | proper#(X) | |
| active#(U32(X1, X2)) | → | active#(X1) | active#(U32(X1, X2)) | → | U32#(active(X1), X2) | |
| proper#(s(X)) | → | s#(proper(X)) | active#(U23(X)) | → | active#(X) | |
| active#(zeros) | → | cons#(0, zeros) | active#(U46(X)) | → | U46#(active(X)) | |
| top#(ok(X)) | → | top#(active(X)) | U61#(ok(X)) | → | U61#(X) | |
| proper#(U31(X1, X2)) | → | proper#(X1) | U71#(ok(X)) | → | U71#(X) | |
| U92#(mark(X1), X2, X3) | → | U92#(X1, X2, X3) | U11#(ok(X1), ok(X2)) | → | U11#(X1, X2) | |
| U23#(mark(X)) | → | U23#(X) | proper#(isNatIList(X)) | → | isNatIList#(proper(X)) | |
| proper#(U83(X1, X2, X3)) | → | proper#(X1) | active#(cons(X1, X2)) | → | cons#(active(X1), X2) | |
| active#(U45(tt, V2)) | → | isNatIList#(V2) | U82#(ok(X1), ok(X2), ok(X3)) | → | U82#(X1, X2, X3) | |
| active#(U81(tt, V1, V2)) | → | isNatKind#(V1) | proper#(U83(X1, X2, X3)) | → | proper#(X2) | |
| U93#(mark(X1), X2, X3) | → | U93#(X1, X2, X3) | active#(U91(X1, X2, X3)) | → | active#(X1) | |
| active#(U11(X1, X2)) | → | active#(X1) | active#(U23(X)) | → | U23#(active(X)) | |
| U81#(mark(X1), X2, X3) | → | U81#(X1, X2, X3) | U84#(mark(X1), X2, X3) | → | U84#(X1, X2, X3) | |
| active#(U82(tt, V1, V2)) | → | isNatIListKind#(V2) | top#(mark(X)) | → | top#(proper(X)) | |
| U42#(mark(X1), X2, X3) | → | U42#(X1, X2, X3) | proper#(cons(X1, X2)) | → | proper#(X2) | |
| proper#(isNatIList(X)) | → | proper#(X) | U13#(ok(X)) | → | U13#(X) | |
| active#(U84(X1, X2, X3)) | → | active#(X1) | active#(U43(X1, X2, X3)) | → | U43#(active(X1), X2, X3) | |
| active#(U45(X1, X2)) | → | active#(X1) | proper#(U22(X1, X2)) | → | proper#(X2) | |
| isNatList#(ok(X)) | → | isNatList#(X) | proper#(U84(X1, X2, X3)) | → | U84#(proper(X1), proper(X2), proper(X3)) | |
| active#(U84(tt, V1, V2)) | → | isNat#(V1) | active#(U92(X1, X2, X3)) | → | U92#(active(X1), X2, X3) | |
| proper#(U82(X1, X2, X3)) | → | U82#(proper(X1), proper(X2), proper(X3)) | active#(U71(X)) | → | U71#(active(X)) | |
| active#(U81(X1, X2, X3)) | → | U81#(active(X1), X2, X3) | proper#(U82(X1, X2, X3)) | → | proper#(X3) | |
| active#(U84(X1, X2, X3)) | → | U84#(active(X1), X2, X3) | proper#(U12(X1, X2)) | → | U12#(proper(X1), proper(X2)) | |
| proper#(U83(X1, X2, X3)) | → | U83#(proper(X1), proper(X2), proper(X3)) | proper#(U41(X1, X2, X3)) | → | proper#(X3) | |
| proper#(U85(X1, X2)) | → | proper#(X1) | U86#(ok(X)) | → | U86#(X) | |
| proper#(U13(X)) | → | proper#(X) | active#(U85(X1, X2)) | → | active#(X1) | |
| proper#(U21(X1, X2)) | → | proper#(X2) | proper#(U22(X1, X2)) | → | proper#(X1) | |
| active#(U31(X1, X2)) | → | U31#(active(X1), X2) | U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | |
| proper#(isNatIListKind(X)) | → | isNatIListKind#(proper(X)) | U23#(ok(X)) | → | U23#(X) | |
| proper#(U86(X)) | → | U86#(proper(X)) | active#(U44(tt, V1, V2)) | → | isNat#(V1) | |
| active#(isNatKind(s(V1))) | → | isNatKind#(V1) | active#(U94(X1, X2)) | → | U94#(active(X1), X2) | |
| proper#(U21(X1, X2)) | → | U21#(proper(X1), proper(X2)) | active#(length(X)) | → | length#(active(X)) | |
| proper#(U43(X1, X2, X3)) | → | U43#(proper(X1), proper(X2), proper(X3)) | active#(U92(tt, L, N)) | → | U93#(isNat(N), L, N) | |
| active#(isNatIList(cons(V1, V2))) | → | isNatKind#(V1) | U46#(mark(X)) | → | U46#(X) | |
| U31#(mark(X1), X2) | → | U31#(X1, X2) | proper#(U92(X1, X2, X3)) | → | proper#(X2) | |
| proper#(U44(X1, X2, X3)) | → | proper#(X3) | active#(U21(X1, X2)) | → | active#(X1) | |
| proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) | proper#(U93(X1, X2, X3)) | → | proper#(X2) | |
| active#(U12(tt, V1)) | → | U13#(isNatList(V1)) | proper#(U82(X1, X2, X3)) | → | proper#(X2) | |
| U21#(mark(X1), X2) | → | U21#(X1, X2) | U32#(mark(X1), X2) | → | U32#(X1, X2) | |
| active#(isNatList(cons(V1, V2))) | → | isNatKind#(V1) | U46#(ok(X)) | → | U46#(X) | |
| active#(U71(X)) | → | active#(X) | proper#(U52(X)) | → | proper#(X) | |
| proper#(U83(X1, X2, X3)) | → | proper#(X3) |
| active(zeros) | → | mark(cons(0, zeros)) | active(U11(tt, V1)) | → | mark(U12(isNatIListKind(V1), V1)) | |
| active(U12(tt, V1)) | → | mark(U13(isNatList(V1))) | active(U13(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, V)) | → | mark(U32(isNatIListKind(V), V)) | |
| active(U32(tt, V)) | → | mark(U33(isNatList(V))) | active(U33(tt)) | → | mark(tt) | |
| active(U41(tt, V1, V2)) | → | mark(U42(isNatKind(V1), V1, V2)) | active(U42(tt, V1, V2)) | → | mark(U43(isNatIListKind(V2), V1, V2)) | |
| active(U43(tt, V1, V2)) | → | mark(U44(isNatIListKind(V2), V1, V2)) | active(U44(tt, V1, V2)) | → | mark(U45(isNat(V1), V2)) | |
| active(U45(tt, V2)) | → | mark(U46(isNatIList(V2))) | active(U46(tt)) | → | mark(tt) | |
| active(U51(tt, V2)) | → | mark(U52(isNatIListKind(V2))) | active(U52(tt)) | → | mark(tt) | |
| active(U61(tt)) | → | mark(tt) | active(U71(tt)) | → | mark(tt) | |
| active(U81(tt, V1, V2)) | → | mark(U82(isNatKind(V1), V1, V2)) | active(U82(tt, V1, V2)) | → | mark(U83(isNatIListKind(V2), V1, V2)) | |
| active(U83(tt, V1, V2)) | → | mark(U84(isNatIListKind(V2), V1, V2)) | active(U84(tt, V1, V2)) | → | mark(U85(isNat(V1), V2)) | |
| active(U85(tt, V2)) | → | mark(U86(isNatList(V2))) | active(U86(tt)) | → | mark(tt) | |
| active(U91(tt, L, N)) | → | mark(U92(isNatIListKind(L), L, N)) | active(U92(tt, L, N)) | → | mark(U93(isNat(N), L, N)) | |
| active(U93(tt, L, N)) | → | mark(U94(isNatKind(N), L)) | active(U94(tt, L)) | → | mark(s(length(L))) | |
| 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(isNatKind(V1), V1, V2)) | |
| active(isNatIListKind(nil)) | → | mark(tt) | active(isNatIListKind(zeros)) | → | mark(tt) | |
| active(isNatIListKind(cons(V1, V2))) | → | mark(U51(isNatKind(V1), V2)) | active(isNatKind(0)) | → | mark(tt) | |
| active(isNatKind(length(V1))) | → | mark(U61(isNatIListKind(V1))) | active(isNatKind(s(V1))) | → | mark(U71(isNatKind(V1))) | |
| active(isNatList(nil)) | → | mark(tt) | active(isNatList(cons(V1, V2))) | → | mark(U81(isNatKind(V1), V1, V2)) | |
| active(length(nil)) | → | mark(0) | active(length(cons(N, L))) | → | mark(U91(isNatList(L), L, N)) | |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | active(U11(X1, X2)) | → | U11(active(X1), X2) | |
| active(U12(X1, X2)) | → | U12(active(X1), X2) | active(U13(X)) | → | U13(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)) | → | U31(active(X1), X2) | |
| active(U32(X1, X2)) | → | U32(active(X1), X2) | active(U33(X)) | → | U33(active(X)) | |
| active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | active(U42(X1, X2, X3)) | → | U42(active(X1), X2, X3) | |
| active(U43(X1, X2, X3)) | → | U43(active(X1), X2, X3) | active(U44(X1, X2, X3)) | → | U44(active(X1), X2, X3) | |
| active(U45(X1, X2)) | → | U45(active(X1), X2) | active(U46(X)) | → | U46(active(X)) | |
| active(U51(X1, X2)) | → | U51(active(X1), X2) | active(U52(X)) | → | U52(active(X)) | |
| active(U61(X)) | → | U61(active(X)) | active(U71(X)) | → | U71(active(X)) | |
| 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(U85(X1, X2)) | → | U85(active(X1), X2) | active(U86(X)) | → | U86(active(X)) | |
| active(U91(X1, X2, X3)) | → | U91(active(X1), X2, X3) | active(U92(X1, X2, X3)) | → | U92(active(X1), X2, X3) | |
| active(U93(X1, X2, X3)) | → | U93(active(X1), X2, X3) | active(U94(X1, X2)) | → | U94(active(X1), X2) | |
| active(s(X)) | → | s(active(X)) | active(length(X)) | → | length(active(X)) | |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | U11(mark(X1), X2) | → | mark(U11(X1, X2)) | |
| U12(mark(X1), X2) | → | mark(U12(X1, X2)) | U13(mark(X)) | → | mark(U13(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) | → | mark(U31(X1, X2)) | |
| U32(mark(X1), X2) | → | mark(U32(X1, X2)) | U33(mark(X)) | → | mark(U33(X)) | |
| U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | U42(mark(X1), X2, X3) | → | mark(U42(X1, X2, X3)) | |
| U43(mark(X1), X2, X3) | → | mark(U43(X1, X2, X3)) | U44(mark(X1), X2, X3) | → | mark(U44(X1, X2, X3)) | |
| U45(mark(X1), X2) | → | mark(U45(X1, X2)) | U46(mark(X)) | → | mark(U46(X)) | |
| U51(mark(X1), X2) | → | mark(U51(X1, X2)) | U52(mark(X)) | → | mark(U52(X)) | |
| U61(mark(X)) | → | mark(U61(X)) | U71(mark(X)) | → | mark(U71(X)) | |
| 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)) | |
| U85(mark(X1), X2) | → | mark(U85(X1, X2)) | U86(mark(X)) | → | mark(U86(X)) | |
| U91(mark(X1), X2, X3) | → | mark(U91(X1, X2, X3)) | U92(mark(X1), X2, X3) | → | mark(U92(X1, X2, X3)) | |
| U93(mark(X1), X2, X3) | → | mark(U93(X1, X2, X3)) | U94(mark(X1), X2) | → | mark(U94(X1, X2)) | |
| s(mark(X)) | → | mark(s(X)) | length(mark(X)) | → | mark(length(X)) | |
| proper(zeros) | → | ok(zeros) | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | |
| proper(0) | → | ok(0) | proper(U11(X1, X2)) | → | U11(proper(X1), proper(X2)) | |
| proper(tt) | → | ok(tt) | proper(U12(X1, X2)) | → | U12(proper(X1), proper(X2)) | |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | proper(U13(X)) | → | U13(proper(X)) | |
| proper(isNatList(X)) | → | isNatList(proper(X)) | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | |
| proper(U22(X1, X2)) | → | U22(proper(X1), proper(X2)) | proper(isNatKind(X)) | → | isNatKind(proper(X)) | |
| proper(U23(X)) | → | U23(proper(X)) | proper(isNat(X)) | → | isNat(proper(X)) | |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | proper(U32(X1, X2)) | → | U32(proper(X1), proper(X2)) | |
| proper(U33(X)) | → | U33(proper(X)) | proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | |
| proper(U42(X1, X2, X3)) | → | U42(proper(X1), proper(X2), proper(X3)) | proper(U43(X1, X2, X3)) | → | U43(proper(X1), proper(X2), proper(X3)) | |
| proper(U44(X1, X2, X3)) | → | U44(proper(X1), proper(X2), proper(X3)) | proper(U45(X1, X2)) | → | U45(proper(X1), proper(X2)) | |
| proper(U46(X)) | → | U46(proper(X)) | proper(isNatIList(X)) | → | isNatIList(proper(X)) | |
| proper(U51(X1, X2)) | → | U51(proper(X1), proper(X2)) | proper(U52(X)) | → | U52(proper(X)) | |
| proper(U61(X)) | → | U61(proper(X)) | proper(U71(X)) | → | U71(proper(X)) | |
| 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(U85(X1, X2)) | → | U85(proper(X1), proper(X2)) | proper(U86(X)) | → | U86(proper(X)) | |
| proper(U91(X1, X2, X3)) | → | U91(proper(X1), proper(X2), proper(X3)) | proper(U92(X1, X2, X3)) | → | U92(proper(X1), proper(X2), proper(X3)) | |
| proper(U93(X1, X2, X3)) | → | U93(proper(X1), proper(X2), proper(X3)) | proper(U94(X1, X2)) | → | U94(proper(X1), proper(X2)) | |
| proper(s(X)) | → | s(proper(X)) | proper(length(X)) | → | length(proper(X)) | |
| proper(nil) | → | ok(nil) | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | |
| U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | U12(ok(X1), ok(X2)) | → | ok(U12(X1, X2)) | |
| isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) | U13(ok(X)) | → | ok(U13(X)) | |
| isNatList(ok(X)) | → | ok(isNatList(X)) | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | |
| U22(ok(X1), ok(X2)) | → | ok(U22(X1, X2)) | isNatKind(ok(X)) | → | ok(isNatKind(X)) | |
| U23(ok(X)) | → | ok(U23(X)) | isNat(ok(X)) | → | ok(isNat(X)) | |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | U32(ok(X1), ok(X2)) | → | ok(U32(X1, X2)) | |
| U33(ok(X)) | → | ok(U33(X)) | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | |
| U42(ok(X1), ok(X2), ok(X3)) | → | ok(U42(X1, X2, X3)) | U43(ok(X1), ok(X2), ok(X3)) | → | ok(U43(X1, X2, X3)) | |
| U44(ok(X1), ok(X2), ok(X3)) | → | ok(U44(X1, X2, X3)) | U45(ok(X1), ok(X2)) | → | ok(U45(X1, X2)) | |
| U46(ok(X)) | → | ok(U46(X)) | isNatIList(ok(X)) | → | ok(isNatIList(X)) | |
| U51(ok(X1), ok(X2)) | → | ok(U51(X1, X2)) | U52(ok(X)) | → | ok(U52(X)) | |
| U61(ok(X)) | → | ok(U61(X)) | U71(ok(X)) | → | ok(U71(X)) | |
| 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)) | |
| U85(ok(X1), ok(X2)) | → | ok(U85(X1, X2)) | U86(ok(X)) | → | ok(U86(X)) | |
| U91(ok(X1), ok(X2), ok(X3)) | → | ok(U91(X1, X2, X3)) | U92(ok(X1), ok(X2), ok(X3)) | → | ok(U92(X1, X2, X3)) | |
| U93(ok(X1), ok(X2), ok(X3)) | → | ok(U93(X1, X2, X3)) | U94(ok(X1), ok(X2)) | → | ok(U94(X1, X2)) | |
| s(ok(X)) | → | ok(s(X)) | length(ok(X)) | → | ok(length(X)) | |
| top(mark(X)) | → | top(proper(X)) | top(ok(X)) | → | top(active(X)) |
Termination of terms over the following signature is verified: isNatIListKind, U94, isNat, U46, U45, U44, U61, U43, U93, U42, U92, U41, U91, length, ok, proper, U23, U21, top, cons, U22, isNatIList, isNatKind, U83, U84, U85, mark, U86, U71, 0, s, U51, isNatList, tt, zeros, U82, U81, U52, U11, active, U12, U13, U31, U32, U33, nil