The TRS could not be proven terminating. The proof attempt took 60023 ms.
Problem 1 remains open; application of the following processors failed [DependencyGraph (60003ms)].
| proper#(cons(X1, X2)) | → | proper#(X1) | proper#(length(X)) | → | length#(proper(X)) | |
| active#(U52(tt, V2)) | → | isNatList#(V2) | active#(U61(tt, V1, V2)) | → | isNat#(V1) | |
| active#(U12(X)) | → | active#(X) | active#(take(s(M), cons(N, IL))) | → | isNatKind#(M) | |
| U11#(mark(X1), X2) | → | U11#(X1, X2) | active#(isNatIListKind(cons(V1, V2))) | → | isNatKind#(V1) | |
| active#(take(s(M), cons(N, IL))) | → | and#(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))) | top#(mark(X)) | → | proper#(X) | |
| U43#(ok(X)) | → | U43#(X) | length#(mark(X)) | → | length#(X) | |
| isNatKind#(ok(X)) | → | isNatKind#(X) | U81#(mark(X)) | → | U81#(X) | |
| proper#(U71(X1, X2)) | → | U71#(proper(X1), proper(X2)) | active#(U32(X)) | → | active#(X) | |
| active#(length(X)) | → | active#(X) | proper#(U12(X)) | → | U12#(proper(X)) | |
| and#(mark(X1), X2) | → | and#(X1, X2) | active#(U21(tt, V1)) | → | isNat#(V1) | |
| U42#(mark(X1), X2) | → | U42#(X1, X2) | active#(U42(X1, X2)) | → | U42#(active(X1), X2) | |
| U53#(mark(X)) | → | U53#(X) | active#(isNatList(take(V1, V2))) | → | U61#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | |
| proper#(U52(X1, X2)) | → | proper#(X1) | proper#(isNatKind(X)) | → | isNatKind#(proper(X)) | |
| active#(U41(X1, X2, X3)) | → | U41#(active(X1), X2, X3) | active#(take(s(M), cons(N, IL))) | → | isNat#(N) | |
| proper#(U62(X1, X2)) | → | proper#(X1) | U32#(mark(X)) | → | U32#(X) | |
| active#(isNatIListKind(take(V1, V2))) | → | isNatKind#(V1) | active#(U52(X1, X2)) | → | active#(X1) | |
| active#(U53(X)) | → | active#(X) | active#(and(X1, X2)) | → | and#(active(X1), X2) | |
| U52#(ok(X1), ok(X2)) | → | U52#(X1, X2) | active#(isNatIListKind(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) | |
| proper#(U41(X1, X2, X3)) | → | proper#(X1) | active#(U31(X1, X2)) | → | active#(X1) | |
| active#(isNatIListKind(take(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) | proper#(isNatList(X)) | → | proper#(X) | |
| U61#(mark(X1), X2, X3) | → | U61#(X1, X2, X3) | take#(ok(X1), ok(X2)) | → | take#(X1, X2) | |
| active#(U41(tt, V1, V2)) | → | U42#(isNat(V1), V2) | U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) | |
| proper#(U63(X)) | → | U63#(proper(X)) | active#(isNatIList(cons(V1, V2))) | → | isNatIListKind#(V2) | |
| active#(U21(X1, X2)) | → | U21#(active(X1), X2) | proper#(isNat(X)) | → | proper#(X) | |
| active#(isNatList(take(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) | active#(length(cons(N, L))) | → | and#(isNatList(L), isNatIListKind(L)) | |
| proper#(U63(X)) | → | proper#(X) | active#(U11(X1, X2)) | → | U11#(active(X1), X2) | |
| proper#(U42(X1, X2)) | → | U42#(proper(X1), proper(X2)) | U91#(mark(X1), X2, X3, X4) | → | U91#(X1, X2, X3, X4) | |
| active#(U51(X1, X2, X3)) | → | U51#(active(X1), X2, X3) | active#(isNat(s(V1))) | → | U21#(isNatKind(V1), V1) | |
| proper#(U31(X1, X2)) | → | U31#(proper(X1), proper(X2)) | proper#(U21(X1, X2)) | → | proper#(X1) | |
| U81#(ok(X)) | → | U81#(X) | proper#(U91(X1, X2, X3, X4)) | → | proper#(X1) | |
| U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) | active#(U11(tt, V1)) | → | isNatList#(V1) | |
| active#(U91(X1, X2, X3, X4)) | → | U91#(active(X1), X2, X3, X4) | U52#(mark(X1), X2) | → | U52#(X1, X2) | |
| active#(U91(tt, IL, M, N)) | → | take#(M, IL) | active#(isNatList(take(V1, V2))) | → | isNatKind#(V1) | |
| proper#(U91(X1, X2, X3, X4)) | → | proper#(X2) | active#(take(s(M), cons(N, IL))) | → | U91#(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N) | |
| active#(take(s(M), cons(N, IL))) | → | isNatIList#(IL) | active#(take(0, IL)) | → | isNatIListKind#(IL) | |
| isNatIListKind#(ok(X)) | → | isNatIListKind#(X) | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) | |
| active#(U42(tt, V2)) | → | isNatIList#(V2) | proper#(U11(X1, X2)) | → | U11#(proper(X1), proper(X2)) | |
| proper#(and(X1, X2)) | → | and#(proper(X1), proper(X2)) | U71#(ok(X1), ok(X2)) | → | U71#(X1, X2) | |
| U12#(mark(X)) | → | U12#(X) | proper#(U71(X1, X2)) | → | proper#(X2) | |
| active#(U62(X1, X2)) | → | active#(X1) | proper#(and(X1, X2)) | → | proper#(X2) | |
| active#(U11(tt, V1)) | → | U12#(isNatList(V1)) | active#(take(s(M), cons(N, IL))) | → | and#(isNat(M), isNatKind(M)) | |
| length#(ok(X)) | → | length#(X) | U22#(mark(X)) | → | U22#(X) | |
| proper#(U41(X1, X2, X3)) | → | U41#(proper(X1), proper(X2), proper(X3)) | active#(take(s(M), cons(N, IL))) | → | isNatKind#(N) | |
| proper#(U43(X)) | → | proper#(X) | active#(U51(X1, X2, X3)) | → | active#(X1) | |
| U22#(ok(X)) | → | U22#(X) | 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)) | |
| take#(X1, mark(X2)) | → | take#(X1, X2) | active#(U81(X)) | → | active#(X) | |
| active#(take(0, IL)) | → | U81#(and(isNatIList(IL), isNatIListKind(IL))) | active#(U21(tt, V1)) | → | U22#(isNat(V1)) | |
| proper#(s(X)) | → | proper#(X) | proper#(isNatList(X)) | → | isNatList#(proper(X)) | |
| active#(isNat(length(V1))) | → | U11#(isNatIListKind(V1), V1) | proper#(take(X1, X2)) | → | take#(proper(X1), proper(X2)) | |
| active#(cons(X1, X2)) | → | active#(X1) | take#(mark(X1), X2) | → | take#(X1, X2) | |
| active#(length(cons(N, L))) | → | isNatKind#(N) | proper#(U31(X1, X2)) | → | proper#(X2) | |
| and#(ok(X1), ok(X2)) | → | and#(X1, X2) | active#(length(cons(N, L))) | → | and#(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))) | |
| active#(length(cons(N, L))) | → | isNatList#(L) | proper#(U32(X)) | → | proper#(X) | |
| active#(U12(X)) | → | U12#(active(X)) | proper#(take(X1, X2)) | → | proper#(X1) | |
| active#(length(cons(N, L))) | → | isNat#(N) | active#(U22(X)) | → | U22#(active(X)) | |
| active#(U71(X1, X2)) | → | active#(X1) | U43#(mark(X)) | → | U43#(X) | |
| U61#(ok(X1), ok(X2), ok(X3)) | → | U61#(X1, X2, X3) | proper#(take(X1, X2)) | → | proper#(X2) | |
| active#(take(X1, X2)) | → | active#(X1) | U63#(ok(X)) | → | U63#(X) | |
| s#(mark(X)) | → | s#(X) | active#(U91(tt, IL, M, N)) | → | cons#(N, take(M, IL)) | |
| U12#(ok(X)) | → | U12#(X) | active#(U22(X)) | → | active#(X) | |
| active#(s(X)) | → | active#(X) | U63#(mark(X)) | → | U63#(X) | |
| active#(isNatIList(V)) | → | U31#(isNatIListKind(V), V) | U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) | |
| active#(take(0, IL)) | → | isNatIList#(IL) | active#(U61(tt, V1, V2)) | → | U62#(isNat(V1), V2) | |
| proper#(U11(X1, X2)) | → | proper#(X1) | active#(isNatIList(V)) | → | isNatIListKind#(V) | |
| proper#(U62(X1, X2)) | → | U62#(proper(X1), proper(X2)) | proper#(U41(X1, X2, X3)) | → | proper#(X2) | |
| active#(U62(tt, V2)) | → | isNatIList#(V2) | active#(take(X1, X2)) | → | take#(active(X1), X2) | |
| proper#(isNatKind(X)) | → | proper#(X) | proper#(U11(X1, X2)) | → | proper#(X2) | |
| active#(length(cons(N, L))) | → | U71#(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L) | active#(U51(tt, V1, V2)) | → | U52#(isNat(V1), V2) | |
| active#(U43(X)) | → | U43#(active(X)) | active#(take(X1, X2)) | → | active#(X2) | |
| proper#(U61(X1, X2, X3)) | → | U61#(proper(X1), proper(X2), proper(X3)) | active#(U52(tt, V2)) | → | U53#(isNatList(V2)) | |
| isNat#(ok(X)) | → | isNat#(X) | isNatIList#(ok(X)) | → | isNatIList#(X) | |
| active#(U41(X1, X2, X3)) | → | active#(X1) | active#(isNat(s(V1))) | → | isNatKind#(V1) | |
| active#(isNatKind(length(V1))) | → | isNatIListKind#(V1) | active#(take(s(M), cons(N, IL))) | → | isNat#(M) | |
| active#(isNatList(cons(V1, V2))) | → | isNatIListKind#(V2) | proper#(U51(X1, X2, X3)) | → | proper#(X1) | |
| proper#(U32(X)) | → | U32#(proper(X)) | active#(U31(tt, V)) | → | U32#(isNatList(V)) | |
| proper#(U22(X)) | → | proper#(X) | proper#(U61(X1, X2, X3)) | → | proper#(X1) | |
| active#(U53(X)) | → | U53#(active(X)) | active#(isNatList(cons(V1, V2))) | → | U51#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | |
| active#(U62(X1, X2)) | → | U62#(active(X1), X2) | active#(U61(X1, X2, X3)) | → | active#(X1) | |
| proper#(U51(X1, X2, X3)) | → | proper#(X2) | U51#(ok(X1), ok(X2), ok(X3)) | → | U51#(X1, X2, X3) | |
| proper#(U62(X1, X2)) | → | proper#(X2) | proper#(isNatIListKind(X)) | → | proper#(X) | |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | active#(U52(X1, X2)) | → | U52#(active(X1), X2) | |
| top#(ok(X)) | → | active#(X) | active#(take(s(M), cons(N, IL))) | → | isNatIListKind#(IL) | |
| active#(isNatList(cons(V1, V2))) | → | and#(isNatKind(V1), isNatIListKind(V2)) | proper#(U42(X1, X2)) | → | proper#(X1) | |
| U62#(ok(X1), ok(X2)) | → | U62#(X1, X2) | active#(U63(X)) | → | active#(X) | |
| U71#(mark(X1), X2) | → | U71#(X1, X2) | proper#(isNat(X)) | → | isNat#(proper(X)) | |
| active#(isNat(length(V1))) | → | isNatIListKind#(V1) | proper#(U81(X)) | → | U81#(proper(X)) | |
| U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | active#(U61(X1, X2, X3)) | → | U61#(active(X1), X2, X3) | |
| proper#(U12(X)) | → | proper#(X) | proper#(U91(X1, X2, X3, X4)) | → | proper#(X3) | |
| proper#(U22(X)) | → | U22#(proper(X)) | active#(s(X)) | → | s#(active(X)) | |
| s#(ok(X)) | → | s#(X) | proper#(length(X)) | → | proper#(X) | |
| active#(take(X1, X2)) | → | take#(X1, active(X2)) | proper#(s(X)) | → | s#(proper(X)) | |
| active#(zeros) | → | cons#(0, zeros) | active#(isNatList(take(V1, V2))) | → | isNatIListKind#(V2) | |
| active#(isNatIListKind(take(V1, V2))) | → | isNatIListKind#(V2) | top#(ok(X)) | → | top#(active(X)) | |
| active#(U51(tt, V1, V2)) | → | isNat#(V1) | proper#(U42(X1, X2)) | → | proper#(X2) | |
| proper#(U31(X1, X2)) | → | proper#(X1) | U32#(ok(X)) | → | U32#(X) | |
| U11#(ok(X1), ok(X2)) | → | U11#(X1, X2) | active#(U71(tt, L)) | → | length#(L) | |
| active#(U81(X)) | → | U81#(active(X)) | proper#(isNatIList(X)) | → | isNatIList#(proper(X)) | |
| proper#(U52(X1, X2)) | → | U52#(proper(X1), proper(X2)) | active#(take(s(M), cons(N, IL))) | → | and#(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))) | |
| active#(cons(X1, X2)) | → | cons#(active(X1), X2) | active#(U42(tt, V2)) | → | U43#(isNatIList(V2)) | |
| U91#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U91#(X1, X2, X3, X4) | U62#(mark(X1), X2) | → | U62#(X1, X2) | |
| active#(U11(X1, X2)) | → | active#(X1) | active#(take(s(M), cons(N, IL))) | → | and#(isNatIList(IL), isNatIListKind(IL)) | |
| top#(mark(X)) | → | top#(proper(X)) | proper#(cons(X1, X2)) | → | proper#(X2) | |
| proper#(isNatIList(X)) | → | proper#(X) | active#(U62(tt, V2)) | → | U63#(isNatIList(V2)) | |
| active#(U91(X1, X2, X3, X4)) | → | active#(X1) | proper#(U61(X1, X2, X3)) | → | proper#(X2) | |
| isNatList#(ok(X)) | → | isNatList#(X) | active#(U63(X)) | → | U63#(active(X)) | |
| active#(U31(tt, V)) | → | isNatList#(V) | active#(length(cons(N, L))) | → | isNatIListKind#(L) | |
| proper#(U91(X1, X2, X3, X4)) | → | proper#(X4) | active#(take(0, IL)) | → | and#(isNatIList(IL), isNatIListKind(IL)) | |
| proper#(U43(X)) | → | U43#(proper(X)) | proper#(U51(X1, X2, X3)) | → | U51#(proper(X1), proper(X2), proper(X3)) | |
| active#(U71(X1, X2)) | → | U71#(active(X1), X2) | active#(take(s(M), cons(N, IL))) | → | and#(isNat(N), isNatKind(N)) | |
| proper#(U53(X)) | → | U53#(proper(X)) | proper#(U61(X1, X2, X3)) | → | proper#(X3) | |
| proper#(U41(X1, X2, X3)) | → | proper#(X3) | active#(U42(X1, X2)) | → | active#(X1) | |
| proper#(and(X1, X2)) | → | proper#(X1) | active#(U43(X)) | → | active#(X) | |
| U53#(ok(X)) | → | U53#(X) | active#(U41(tt, V1, V2)) | → | isNat#(V1) | |
| proper#(U21(X1, X2)) | → | proper#(X2) | active#(U31(X1, X2)) | → | U31#(active(X1), X2) | |
| proper#(U71(X1, X2)) | → | proper#(X1) | proper#(isNatIListKind(X)) | → | isNatIListKind#(proper(X)) | |
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | proper#(U51(X1, X2, X3)) | → | proper#(X3) | |
| active#(U71(tt, L)) | → | s#(length(L)) | active#(isNatKind(s(V1))) | → | isNatKind#(V1) | |
| proper#(U21(X1, X2)) | → | U21#(proper(X1), proper(X2)) | proper#(U53(X)) | → | proper#(X) | |
| active#(length(X)) | → | length#(active(X)) | U42#(ok(X1), ok(X2)) | → | U42#(X1, X2) | |
| active#(U32(X)) | → | U32#(active(X)) | active#(isNatIList(cons(V1, V2))) | → | isNatKind#(V1) | |
| proper#(U91(X1, X2, X3, X4)) | → | U91#(proper(X1), proper(X2), proper(X3), proper(X4)) | proper#(U81(X)) | → | proper#(X) | |
| U31#(mark(X1), X2) | → | U31#(X1, X2) | proper#(U52(X1, X2)) | → | proper#(X2) | |
| active#(U21(X1, X2)) | → | active#(X1) | active#(isNatIList(cons(V1, V2))) | → | U41#(and(isNatKind(V1), isNatIListKind(V2)), V1, V2) | |
| proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) | U21#(mark(X1), X2) | → | U21#(X1, X2) | |
| active#(isNatList(cons(V1, V2))) | → | isNatKind#(V1) | active#(and(X1, X2)) | → | active#(X1) |
| 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, V1, V2)) | → | mark(U62(isNat(V1), V2)) | |
| active(U62(tt, V2)) | → | mark(U63(isNatIList(V2))) | active(U63(tt)) | → | mark(tt) | |
| active(U71(tt, L)) | → | mark(s(length(L))) | active(U81(tt)) | → | mark(nil) | |
| active(U91(tt, IL, M, N)) | → | mark(cons(N, take(M, IL))) | 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(isNatIListKind(take(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(isNatList(take(V1, V2))) | → | mark(U61(and(isNatKind(V1), isNatIListKind(V2)), V1, V2)) | |
| active(length(nil)) | → | mark(0) | active(length(cons(N, L))) | → | mark(U71(and(and(isNatList(L), isNatIListKind(L)), and(isNat(N), isNatKind(N))), L)) | |
| active(take(0, IL)) | → | mark(U81(and(isNatIList(IL), isNatIListKind(IL)))) | active(take(s(M), cons(N, IL))) | → | mark(U91(and(and(isNatIList(IL), isNatIListKind(IL)), and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))), IL, M, N)) | |
| active(cons(X1, X2)) | → | cons(active(X1), X2) | active(U11(X1, X2)) | → | U11(active(X1), X2) | |
| active(U12(X)) | → | U12(active(X)) | active(U21(X1, X2)) | → | U21(active(X1), X2) | |
| active(U22(X)) | → | U22(active(X)) | active(U31(X1, X2)) | → | U31(active(X1), X2) | |
| active(U32(X)) | → | U32(active(X)) | active(U41(X1, X2, X3)) | → | U41(active(X1), X2, X3) | |
| active(U42(X1, X2)) | → | U42(active(X1), X2) | active(U43(X)) | → | U43(active(X)) | |
| active(U51(X1, X2, X3)) | → | U51(active(X1), X2, X3) | active(U52(X1, X2)) | → | U52(active(X1), X2) | |
| active(U53(X)) | → | U53(active(X)) | active(U61(X1, X2, X3)) | → | U61(active(X1), X2, X3) | |
| active(U62(X1, X2)) | → | U62(active(X1), X2) | active(U63(X)) | → | U63(active(X)) | |
| active(U71(X1, X2)) | → | U71(active(X1), X2) | active(s(X)) | → | s(active(X)) | |
| active(length(X)) | → | length(active(X)) | active(U81(X)) | → | U81(active(X)) | |
| active(U91(X1, X2, X3, X4)) | → | U91(active(X1), X2, X3, X4) | active(take(X1, X2)) | → | take(active(X1), X2) | |
| active(take(X1, X2)) | → | take(X1, active(X2)) | active(and(X1, X2)) | → | and(active(X1), X2) | |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | U11(mark(X1), X2) | → | mark(U11(X1, X2)) | |
| U12(mark(X)) | → | mark(U12(X)) | U21(mark(X1), X2) | → | mark(U21(X1, X2)) | |
| U22(mark(X)) | → | mark(U22(X)) | U31(mark(X1), X2) | → | mark(U31(X1, X2)) | |
| U32(mark(X)) | → | mark(U32(X)) | U41(mark(X1), X2, X3) | → | mark(U41(X1, X2, X3)) | |
| U42(mark(X1), X2) | → | mark(U42(X1, X2)) | U43(mark(X)) | → | mark(U43(X)) | |
| U51(mark(X1), X2, X3) | → | mark(U51(X1, X2, X3)) | U52(mark(X1), X2) | → | mark(U52(X1, X2)) | |
| U53(mark(X)) | → | mark(U53(X)) | U61(mark(X1), X2, X3) | → | mark(U61(X1, X2, X3)) | |
| U62(mark(X1), X2) | → | mark(U62(X1, X2)) | U63(mark(X)) | → | mark(U63(X)) | |
| U71(mark(X1), X2) | → | mark(U71(X1, X2)) | s(mark(X)) | → | mark(s(X)) | |
| length(mark(X)) | → | mark(length(X)) | U81(mark(X)) | → | mark(U81(X)) | |
| U91(mark(X1), X2, X3, X4) | → | mark(U91(X1, X2, X3, X4)) | take(mark(X1), X2) | → | mark(take(X1, X2)) | |
| take(X1, mark(X2)) | → | mark(take(X1, X2)) | and(mark(X1), X2) | → | mark(and(X1, X2)) | |
| 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(X)) | → | U12(proper(X)) | |
| proper(isNatList(X)) | → | isNatList(proper(X)) | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | |
| proper(U22(X)) | → | U22(proper(X)) | proper(isNat(X)) | → | isNat(proper(X)) | |
| proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | proper(U32(X)) | → | U32(proper(X)) | |
| proper(U41(X1, X2, X3)) | → | U41(proper(X1), proper(X2), proper(X3)) | proper(U42(X1, X2)) | → | U42(proper(X1), proper(X2)) | |
| proper(U43(X)) | → | U43(proper(X)) | proper(isNatIList(X)) | → | isNatIList(proper(X)) | |
| proper(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | proper(U52(X1, X2)) | → | U52(proper(X1), proper(X2)) | |
| proper(U53(X)) | → | U53(proper(X)) | proper(U61(X1, X2, X3)) | → | U61(proper(X1), proper(X2), proper(X3)) | |
| proper(U62(X1, X2)) | → | U62(proper(X1), proper(X2)) | proper(U63(X)) | → | U63(proper(X)) | |
| proper(U71(X1, X2)) | → | U71(proper(X1), proper(X2)) | proper(s(X)) | → | s(proper(X)) | |
| proper(length(X)) | → | length(proper(X)) | proper(U81(X)) | → | U81(proper(X)) | |
| proper(nil) | → | ok(nil) | proper(U91(X1, X2, X3, X4)) | → | U91(proper(X1), proper(X2), proper(X3), proper(X4)) | |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) | |
| proper(isNatIListKind(X)) | → | isNatIListKind(proper(X)) | proper(isNatKind(X)) | → | isNatKind(proper(X)) | |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | U11(ok(X1), ok(X2)) | → | ok(U11(X1, X2)) | |
| U12(ok(X)) | → | ok(U12(X)) | isNatList(ok(X)) | → | ok(isNatList(X)) | |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | U22(ok(X)) | → | ok(U22(X)) | |
| isNat(ok(X)) | → | ok(isNat(X)) | U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | |
| U32(ok(X)) | → | ok(U32(X)) | U41(ok(X1), ok(X2), ok(X3)) | → | ok(U41(X1, X2, X3)) | |
| U42(ok(X1), ok(X2)) | → | ok(U42(X1, X2)) | U43(ok(X)) | → | ok(U43(X)) | |
| isNatIList(ok(X)) | → | ok(isNatIList(X)) | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | |
| U52(ok(X1), ok(X2)) | → | ok(U52(X1, X2)) | U53(ok(X)) | → | ok(U53(X)) | |
| U61(ok(X1), ok(X2), ok(X3)) | → | ok(U61(X1, X2, X3)) | U62(ok(X1), ok(X2)) | → | ok(U62(X1, X2)) | |
| U63(ok(X)) | → | ok(U63(X)) | U71(ok(X1), ok(X2)) | → | ok(U71(X1, X2)) | |
| s(ok(X)) | → | ok(s(X)) | length(ok(X)) | → | ok(length(X)) | |
| U81(ok(X)) | → | ok(U81(X)) | U91(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U91(X1, X2, X3, X4)) | |
| take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | |
| isNatIListKind(ok(X)) | → | ok(isNatIListKind(X)) | isNatKind(ok(X)) | → | ok(isNatKind(X)) | |
| top(mark(X)) | → | top(proper(X)) | top(ok(X)) | → | top(active(X)) |
Termination of terms over the following signature is verified: isNatIListKind, isNat, U63, U62, U43, U61, U42, U41, U91, length, ok, proper, U21, top, U22, cons, isNatIList, isNatKind, mark, and, U71, 0, isNatList, U51, s, tt, zeros, take, U53, U81, U52, U11, active, U12, U31, U32, nil