The TRS could not be proven terminating. The proof attempt took 60002 ms.
Problem 1 remains open; application of the following processors failed [DependencyGraph (timeout)].
| active#(isLNat(afterNth(V1, V2))) | → | isNaturalKind#(V1) | proper#(U92(X)) | → | U92#(proper(X)) | |
| proper#(cons(X1, X2)) | → | proper#(X1) | U103#(ok(X)) | → | U103#(X) | |
| proper#(tail(X)) | → | proper#(X) | proper#(U91(X1, X2)) | → | proper#(X2) | |
| proper#(U161(X1, X2)) | → | proper#(X2) | active#(isNatural(sel(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | |
| active#(U191(tt, XS)) | → | pair#(nil, XS) | active#(U121(tt, V1)) | → | U122#(isNatural(V1)) | |
| active#(U142(tt, V2)) | → | U143#(isLNat(V2)) | active#(U202(pair(YS, ZS), X)) | → | cons#(X, YS) | |
| active#(isLNat(cons(V1, V2))) | → | isNaturalKind#(V1) | isNatural#(ok(X)) | → | isNatural#(X) | |
| active#(isLNatKind(fst(V1))) | → | isPLNatKind#(V1) | proper#(U11(X1, X2, X3)) | → | proper#(X2) | |
| U81#(ok(X1), ok(X2)) | → | U81#(X1, X2) | proper#(U132(X1, X2)) | → | proper#(X1) | |
| U43#(ok(X)) | → | U43#(X) | active#(U102(X1, X2)) | → | active#(X1) | |
| active#(U11(tt, N, XS)) | → | splitAt#(N, XS) | active#(isLNat(take(V1, V2))) | → | isNaturalKind#(V1) | |
| U141#(mark(X1), X2, X3) | → | U141#(X1, X2, X3) | proper#(U211(X1, X2)) | → | proper#(X1) | |
| splitAt#(X1, mark(X2)) | → | splitAt#(X1, X2) | active#(afterNth(X1, X2)) | → | afterNth#(active(X1), X2) | |
| U42#(mark(X1), X2) | → | U42#(X1, X2) | active#(head(cons(N, XS))) | → | isLNatKind#(XS) | |
| active#(U42(X1, X2)) | → | U42#(active(X1), X2) | active#(isLNatKind(afterNth(V1, V2))) | → | isLNatKind#(V2) | |
| active#(fst(pair(X, Y))) | → | isLNatKind#(X) | U53#(mark(X)) | → | U53#(X) | |
| proper#(isPLNatKind(X)) | → | proper#(X) | active#(isLNat(cons(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | |
| active#(isLNat(tail(V1))) | → | U91#(isLNatKind(V1), V1) | active#(U121(X1, X2)) | → | active#(X1) | |
| proper#(isPLNatKind(X)) | → | isPLNatKind#(proper(X)) | active#(U131(X1, X2, X3)) | → | U131#(active(X1), X2, X3) | |
| natsFrom#(mark(X)) | → | natsFrom#(X) | active#(splitAt(s(N), cons(X, XS))) | → | and#(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))) | |
| active#(U91(tt, V1)) | → | isLNat#(V1) | active#(take(N, XS)) | → | isNatural#(N) | |
| active#(splitAt(0, XS)) | → | isLNat#(XS) | active#(isPLNatKind(splitAt(V1, V2))) | → | isLNatKind#(V2) | |
| active#(U53(X)) | → | active#(X) | active#(and(X1, X2)) | → | and#(active(X1), X2) | |
| active#(splitAt(s(N), cons(X, XS))) | → | isLNat#(XS) | active#(isNatural(head(V1))) | → | U111#(isLNatKind(V1), V1) | |
| active#(U81(tt, V1)) | → | isPLNat#(V1) | active#(snd(X)) | → | snd#(active(X)) | |
| active#(take(N, XS)) | → | and#(isLNat(XS), isLNatKind(XS)) | proper#(U81(X1, X2)) | → | U81#(proper(X1), proper(X2)) | |
| proper#(isLNatKind(X)) | → | isLNatKind#(proper(X)) | U191#(mark(X1), X2) | → | U191#(X1, X2) | |
| U41#(ok(X1), ok(X2), ok(X3)) | → | U41#(X1, X2, X3) | active#(U133(X)) | → | U133#(active(X)) | |
| isLNat#(ok(X)) | → | isLNat#(X) | active#(U21(X1, X2)) | → | U21#(active(X1), X2) | |
| active#(sel(X1, X2)) | → | sel#(X1, active(X2)) | proper#(U143(X)) | → | U143#(proper(X)) | |
| active#(snd(pair(X, Y))) | → | isLNatKind#(Y) | U142#(ok(X1), ok(X2)) | → | U142#(X1, X2) | |
| active#(U42(tt, V2)) | → | isLNat#(V2) | U91#(ok(X1), ok(X2)) | → | U91#(X1, X2) | |
| proper#(U171(X1, X2, X3)) | → | proper#(X2) | proper#(U121(X1, X2)) | → | proper#(X1) | |
| active#(afterNth(X1, X2)) | → | active#(X2) | active#(U112(X)) | → | U112#(active(X)) | |
| active#(sel(N, XS)) | → | and#(isLNat(XS), isLNatKind(XS)) | active#(U91(tt, V1)) | → | U92#(isLNat(V1)) | |
| proper#(pair(X1, X2)) | → | pair#(proper(X1), proper(X2)) | proper#(U142(X1, X2)) | → | U142#(proper(X1), proper(X2)) | |
| active#(afterNth(N, XS)) | → | U11#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) | proper#(tail(X)) | → | tail#(proper(X)) | |
| active#(U141(X1, X2, X3)) | → | active#(X1) | active#(U161(X1, X2)) | → | active#(X1) | |
| active#(U161(X1, X2)) | → | U161#(active(X1), X2) | proper#(U21(X1, X2)) | → | proper#(X1) | |
| active#(isLNatKind(take(V1, V2))) | → | isNaturalKind#(V1) | U102#(ok(X1), ok(X2)) | → | U102#(X1, X2) | |
| active#(U82(X)) | → | active#(X) | proper#(U102(X1, X2)) | → | proper#(X1) | |
| proper#(U181(X1, X2)) | → | proper#(X2) | active#(U152(tt, V2)) | → | isLNat#(V2) | |
| U52#(mark(X1), X2) | → | U52#(X1, X2) | active#(U61(tt, V1)) | → | U62#(isPLNat(V1)) | |
| afterNth#(X1, mark(X2)) | → | afterNth#(X1, X2) | proper#(U92(X)) | → | proper#(X) | |
| cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) | active#(sel(N, XS)) | → | and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))) | |
| U71#(ok(X1), ok(X2)) | → | U71#(X1, X2) | proper#(U201(X1, X2, X3, X4)) | → | proper#(X2) | |
| active#(U131(tt, V1, V2)) | → | U132#(isNatural(V1), V2) | active#(afterNth(N, XS)) | → | and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))) | |
| active#(head(cons(N, XS))) | → | isNatural#(N) | proper#(isLNat(X)) | → | isLNat#(proper(X)) | |
| U122#(ok(X)) | → | U122#(X) | active#(sel(X1, X2)) | → | active#(X2) | |
| active#(U52(tt, V2)) | → | isLNat#(V2) | U91#(mark(X1), X2) | → | U91#(X1, X2) | |
| U61#(mark(X1), X2) | → | U61#(X1, X2) | active#(isLNat(natsFrom(V1))) | → | U71#(isNaturalKind(V1), V1) | |
| active#(U101(X1, X2, X3)) | → | active#(X1) | active#(afterNth(N, XS)) | → | and#(isNatural(N), isNaturalKind(N)) | |
| active#(U132(X1, X2)) | → | active#(X1) | proper#(U121(X1, X2)) | → | U121#(proper(X1), proper(X2)) | |
| proper#(U41(X1, X2, X3)) | → | U41#(proper(X1), proper(X2), proper(X3)) | proper#(afterNth(X1, X2)) | → | proper#(X1) | |
| U61#(ok(X1), ok(X2)) | → | U61#(X1, X2) | proper#(U43(X)) | → | proper#(X) | |
| active#(splitAt(s(N), cons(X, XS))) | → | isNatural#(N) | active#(U51(X1, X2, X3)) | → | active#(X1) | |
| proper#(fst(X)) | → | fst#(proper(X)) | active#(isNatural(s(V1))) | → | isNaturalKind#(V1) | |
| active#(U42(tt, V2)) | → | U43#(isLNat(V2)) | proper#(pair(X1, X2)) | → | proper#(X2) | |
| active#(natsFrom(N)) | → | isNatural#(N) | take#(X1, mark(X2)) | → | take#(X1, X2) | |
| active#(U72(X)) | → | U72#(active(X)) | active#(head(cons(N, XS))) | → | and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))) | |
| U191#(ok(X1), ok(X2)) | → | U191#(X1, X2) | active#(isLNatKind(snd(V1))) | → | isPLNatKind#(V1) | |
| active#(U111(tt, V1)) | → | U112#(isLNat(V1)) | proper#(take(X1, X2)) | → | take#(proper(X1), proper(X2)) | |
| active#(U61(tt, V1)) | → | isPLNat#(V1) | active#(snd(X)) | → | active#(X) | |
| active#(cons(X1, X2)) | → | active#(X1) | sel#(mark(X1), X2) | → | sel#(X1, X2) | |
| proper#(U31(X1, X2)) | → | proper#(X2) | active#(take(N, XS)) | → | isLNat#(XS) | |
| active#(pair(X1, X2)) | → | pair#(X1, active(X2)) | and#(ok(X1), ok(X2)) | → | and#(X1, X2) | |
| active#(U11(X1, X2, X3)) | → | U11#(active(X1), X2, X3) | active#(natsFrom(N)) | → | and#(isNatural(N), isNaturalKind(N)) | |
| active#(splitAt(s(N), cons(X, XS))) | → | isLNatKind#(XS) | splitAt#(ok(X1), ok(X2)) | → | splitAt#(X1, X2) | |
| U103#(mark(X)) | → | U103#(X) | proper#(take(X1, X2)) | → | proper#(X1) | |
| active#(U171(tt, N, XS)) | → | head#(afterNth(N, XS)) | proper#(U201(X1, X2, X3, X4)) | → | proper#(X3) | |
| U151#(ok(X1), ok(X2), ok(X3)) | → | U151#(X1, X2, X3) | proper#(U91(X1, X2)) | → | U91#(proper(X1), proper(X2)) | |
| U43#(mark(X)) | → | U43#(X) | active#(sel(N, XS)) | → | isLNatKind#(XS) | |
| U152#(mark(X1), X2) | → | U152#(X1, X2) | active#(isLNat(afterNth(V1, V2))) | → | U41#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2) | |
| proper#(take(X1, X2)) | → | proper#(X2) | proper#(U171(X1, X2, X3)) | → | U171#(proper(X1), proper(X2), proper(X3)) | |
| active#(take(X1, X2)) | → | active#(X1) | active#(U81(X1, X2)) | → | U81#(active(X1), X2) | |
| active#(U151(X1, X2, X3)) | → | active#(X1) | active#(U91(X1, X2)) | → | U91#(active(X1), X2) | |
| s#(mark(X)) | → | s#(X) | active#(tail(cons(N, XS))) | → | isLNat#(XS) | |
| active#(fst(pair(X, Y))) | → | isLNat#(Y) | active#(s(X)) | → | active#(X) | |
| active#(U112(X)) | → | active#(X) | active#(tail(X)) | → | active#(X) | |
| active#(isLNatKind(cons(V1, V2))) | → | isNaturalKind#(V1) | active#(afterNth(N, XS)) | → | isLNatKind#(XS) | |
| U122#(mark(X)) | → | U122#(X) | U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) | |
| active#(sel(N, XS)) | → | isNaturalKind#(N) | active#(isLNat(natsFrom(V1))) | → | isNaturalKind#(V1) | |
| proper#(isNatural(X)) | → | proper#(X) | proper#(U41(X1, X2, X3)) | → | proper#(X2) | |
| active#(U152(X1, X2)) | → | U152#(active(X1), X2) | splitAt#(mark(X1), X2) | → | splitAt#(X1, X2) | |
| active#(U161(tt, N)) | → | natsFrom#(s(N)) | active#(take(X1, X2)) | → | take#(active(X1), X2) | |
| active#(take(N, XS)) | → | isLNatKind#(XS) | active#(U82(X)) | → | U82#(active(X)) | |
| active#(splitAt(X1, X2)) | → | splitAt#(active(X1), X2) | active#(splitAt(X1, X2)) | → | splitAt#(X1, active(X2)) | |
| active#(isLNatKind(afterNth(V1, V2))) | → | isNaturalKind#(V1) | proper#(U201(X1, X2, X3, X4)) | → | proper#(X4) | |
| proper#(U81(X1, X2)) | → | proper#(X2) | active#(U103(X)) | → | active#(X) | |
| active#(isLNat(take(V1, V2))) | → | isLNatKind#(V2) | U132#(ok(X1), ok(X2)) | → | U132#(X1, X2) | |
| active#(U41(tt, V1, V2)) | → | U42#(isNatural(V1), V2) | active#(isLNat(snd(V1))) | → | isPLNatKind#(V1) | |
| active#(U43(X)) | → | U43#(active(X)) | active#(take(X1, X2)) | → | active#(X2) | |
| active#(tail(cons(N, XS))) | → | isNatural#(N) | U112#(mark(X)) | → | U112#(X) | |
| pair#(X1, mark(X2)) | → | pair#(X1, X2) | active#(U62(X)) | → | active#(X) | |
| U111#(mark(X1), X2) | → | U111#(X1, X2) | active#(head(cons(N, XS))) | → | and#(isLNat(XS), isLNatKind(XS)) | |
| U133#(mark(X)) | → | U133#(X) | active#(U41(X1, X2, X3)) | → | active#(X1) | |
| proper#(U51(X1, X2, X3)) | → | proper#(X1) | active#(isPLNatKind(pair(V1, V2))) | → | and#(isLNatKind(V1), isLNatKind(V2)) | |
| U211#(mark(X1), X2) | → | U211#(X1, X2) | proper#(U101(X1, X2, X3)) | → | proper#(X3) | |
| U152#(ok(X1), ok(X2)) | → | U152#(X1, X2) | proper#(U51(X1, X2, X3)) | → | proper#(X2) | |
| active#(take(N, XS)) | → | and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))) | proper#(isNatural(X)) | → | isNatural#(proper(X)) | |
| active#(U52(X1, X2)) | → | U52#(active(X1), X2) | active#(U92(X)) | → | U92#(active(X)) | |
| proper#(U201(X1, X2, X3, X4)) | → | U201#(proper(X1), proper(X2), proper(X3), proper(X4)) | active#(afterNth(X1, X2)) | → | afterNth#(X1, active(X2)) | |
| top#(ok(X)) | → | active#(X) | proper#(U42(X1, X2)) | → | proper#(X1) | |
| active#(head(cons(N, XS))) | → | and#(isNatural(N), isNaturalKind(N)) | active#(isLNat(fst(V1))) | → | U61#(isPLNatKind(V1), V1) | |
| active#(splitAt(s(N), cons(X, XS))) | → | isNaturalKind#(X) | proper#(U152(X1, X2)) | → | proper#(X1) | |
| proper#(U91(X1, X2)) | → | proper#(X1) | proper#(U142(X1, X2)) | → | proper#(X1) | |
| active#(pair(X1, X2)) | → | active#(X1) | active#(U132(tt, V2)) | → | isLNat#(V2) | |
| active#(isLNat(afterNth(V1, V2))) | → | isLNatKind#(V2) | active#(splitAt(0, XS)) | → | U191#(and(isLNat(XS), isLNatKind(XS)), XS) | |
| proper#(U202(X1, X2)) | → | proper#(X2) | proper#(isLNat(X)) | → | proper#(X) | |
| active#(fst(pair(X, Y))) | → | U21#(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X) | active#(U161(tt, N)) | → | cons#(N, natsFrom(s(N))) | |
| pair#(ok(X1), ok(X2)) | → | pair#(X1, X2) | active#(isNatural(sel(V1, V2))) | → | U131#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2) | |
| active#(head(X)) | → | head#(active(X)) | active#(snd(pair(X, Y))) | → | isLNat#(Y) | |
| U201#(mark(X1), X2, X3, X4) | → | U201#(X1, X2, X3, X4) | active#(take(N, XS)) | → | isNaturalKind#(N) | |
| proper#(U11(X1, X2, X3)) | → | proper#(X1) | proper#(U103(X)) | → | U103#(proper(X)) | |
| U112#(ok(X)) | → | U112#(X) | U121#(ok(X1), ok(X2)) | → | U121#(X1, X2) | |
| proper#(s(X)) | → | s#(proper(X)) | active#(splitAt(s(N), cons(X, XS))) | → | and#(isLNat(XS), isLNatKind(XS)) | |
| active#(splitAt(X1, X2)) | → | active#(X1) | top#(ok(X)) | → | top#(active(X)) | |
| active#(U151(tt, V1, V2)) | → | isNatural#(V1) | proper#(U42(X1, X2)) | → | proper#(X2) | |
| proper#(U52(X1, X2)) | → | U52#(proper(X1), proper(X2)) | proper#(U171(X1, X2, X3)) | → | proper#(X1) | |
| active#(U81(X1, X2)) | → | active#(X1) | active#(isPLNatKind(pair(V1, V2))) | → | isLNatKind#(V1) | |
| active#(isPLNat(pair(V1, V2))) | → | U141#(and(isLNatKind(V1), isLNatKind(V2)), V1, V2) | active#(pair(X1, X2)) | → | active#(X2) | |
| snd#(ok(X)) | → | snd#(X) | proper#(U102(X1, X2)) | → | U102#(proper(X1), proper(X2)) | |
| active#(take(N, XS)) | → | U221#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) | active#(isPLNatKind(splitAt(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | |
| active#(head(X)) | → | active#(X) | proper#(U111(X1, X2)) | → | proper#(X1) | |
| active#(U153(X)) | → | active#(X) | active#(U81(tt, V1)) | → | U82#(isPLNat(V1)) | |
| active#(U111(tt, V1)) | → | isLNat#(V1) | proper#(U151(X1, X2, X3)) | → | U151#(proper(X1), proper(X2), proper(X3)) | |
| active#(U103(X)) | → | U103#(active(X)) | active#(U221(tt, N, XS)) | → | splitAt#(N, XS) | |
| proper#(isNaturalKind(X)) | → | proper#(X) | U62#(ok(X)) | → | U62#(X) | |
| active#(U171(X1, X2, X3)) | → | active#(X1) | active#(U11(X1, X2, X3)) | → | active#(X1) | |
| proper#(U152(X1, X2)) | → | U152#(proper(X1), proper(X2)) | U181#(mark(X1), X2) | → | U181#(X1, X2) | |
| active#(fst(pair(X, Y))) | → | and#(isLNat(X), isLNatKind(X)) | proper#(U181(X1, X2)) | → | proper#(X1) | |
| active#(U221(X1, X2, X3)) | → | active#(X1) | proper#(U61(X1, X2)) | → | U61#(proper(X1), proper(X2)) | |
| U92#(ok(X)) | → | U92#(X) | active#(splitAt(s(N), cons(X, XS))) | → | and#(isNatural(X), isNaturalKind(X)) | |
| proper#(U41(X1, X2, X3)) | → | proper#(X3) | active#(isLNat(tail(V1))) | → | isLNatKind#(V1) | |
| proper#(and(X1, X2)) | → | proper#(X1) | proper#(fst(X)) | → | proper#(X) | |
| U53#(ok(X)) | → | U53#(X) | afterNth#(ok(X1), ok(X2)) | → | afterNth#(X1, X2) | |
| active#(isPLNatKind(pair(V1, V2))) | → | isLNatKind#(V2) | U202#(mark(X1), X2) | → | U202#(X1, X2) | |
| proper#(U21(X1, X2)) | → | proper#(X2) | proper#(U122(X)) | → | proper#(X) | |
| active#(splitAt(X1, X2)) | → | active#(X2) | active#(U31(X1, X2)) | → | U31#(active(X1), X2) | |
| proper#(U71(X1, X2)) | → | proper#(X1) | active#(U211(X1, X2)) | → | active#(X1) | |
| U41#(mark(X1), X2, X3) | → | U41#(X1, X2, X3) | proper#(U51(X1, X2, X3)) | → | proper#(X3) | |
| active#(isLNat(take(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | isLNatKind#(ok(X)) | → | isLNatKind#(X) | |
| active#(U181(X1, X2)) | → | active#(X1) | active#(sel(N, XS)) | → | isLNat#(XS) | |
| proper#(U131(X1, X2, X3)) | → | proper#(X3) | active#(isLNat(fst(V1))) | → | isPLNatKind#(V1) | |
| proper#(U133(X)) | → | U133#(proper(X)) | proper#(U53(X)) | → | proper#(X) | |
| proper#(U21(X1, X2)) | → | U21#(proper(X1), proper(X2)) | active#(take(N, XS)) | → | and#(isNatural(N), isNaturalKind(N)) | |
| U81#(mark(X1), X2) | → | U81#(X1, X2) | active#(U51(tt, V1, V2)) | → | U52#(isNatural(V1), V2) | |
| U42#(ok(X1), ok(X2)) | → | U42#(X1, X2) | active#(isNatural(sel(V1, V2))) | → | isLNatKind#(V2) | |
| proper#(U82(X)) | → | proper#(X) | active#(U141(X1, X2, X3)) | → | U141#(active(X1), X2, X3) | |
| active#(isPLNat(pair(V1, V2))) | → | isLNatKind#(V2) | proper#(U221(X1, X2, X3)) | → | proper#(X3) | |
| active#(isPLNatKind(splitAt(V1, V2))) | → | isNaturalKind#(V1) | U21#(mark(X1), X2) | → | U21#(X1, X2) | |
| proper#(U131(X1, X2, X3)) | → | proper#(X1) | active#(U221(tt, N, XS)) | → | fst#(splitAt(N, XS)) | |
| active#(tail(cons(N, XS))) | → | and#(isLNat(XS), isLNatKind(XS)) | active#(and(X1, X2)) | → | active#(X1) | |
| proper#(U171(X1, X2, X3)) | → | proper#(X3) | active#(sel(N, XS)) | → | U171#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS) | |
| proper#(U11(X1, X2, X3)) | → | proper#(X3) | U132#(mark(X1), X2) | → | U132#(X1, X2) | |
| active#(head(cons(N, XS))) | → | U31#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N) | active#(U221(X1, X2, X3)) | → | U221#(active(X1), X2, X3) | |
| U82#(mark(X)) | → | U82#(X) | active#(U52(tt, V2)) | → | U53#(isLNat(V2)) | |
| active#(U211(X1, X2)) | → | U211#(active(X1), X2) | tail#(ok(X)) | → | tail#(X) | |
| proper#(U62(X)) | → | proper#(X) | active#(U122(X)) | → | U122#(active(X)) | |
| active#(U142(X1, X2)) | → | active#(X1) | active#(splitAt(s(N), cons(X, XS))) | → | isNaturalKind#(N) | |
| active#(U141(tt, V1, V2)) | → | isLNat#(V1) | top#(mark(X)) | → | proper#(X) | |
| active#(natsFrom(N)) | → | isNaturalKind#(N) | proper#(U153(X)) | → | U153#(proper(X)) | |
| active#(U101(X1, X2, X3)) | → | U101#(active(X1), X2, X3) | proper#(U71(X1, X2)) | → | U71#(proper(X1), proper(X2)) | |
| active#(U143(X)) | → | active#(X) | active#(U61(X1, X2)) | → | active#(X1) | |
| proper#(isNaturalKind(X)) | → | isNaturalKind#(proper(X)) | and#(mark(X1), X2) | → | and#(X1, X2) | |
| active#(U71(tt, V1)) | → | isNatural#(V1) | active#(U101(tt, V1, V2)) | → | U102#(isNatural(V1), V2) | |
| active#(U161(tt, N)) | → | s#(N) | sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) | |
| active#(fst(pair(X, Y))) | → | and#(isLNat(Y), isLNatKind(Y)) | proper#(U151(X1, X2, X3)) | → | proper#(X3) | |
| proper#(U101(X1, X2, X3)) | → | proper#(X2) | proper#(splitAt(X1, X2)) | → | proper#(X2) | |
| U202#(ok(X1), ok(X2)) | → | U202#(X1, X2) | active#(U151(X1, X2, X3)) | → | U151#(active(X1), X2, X3) | |
| proper#(U52(X1, X2)) | → | proper#(X1) | proper#(head(X)) | → | proper#(X) | |
| proper#(U111(X1, X2)) | → | U111#(proper(X1), proper(X2)) | active#(splitAt(s(N), cons(X, XS))) | → | and#(isNatural(N), isNaturalKind(N)) | |
| active#(U41(X1, X2, X3)) | → | U41#(active(X1), X2, X3) | U102#(mark(X1), X2) | → | U102#(X1, X2) | |
| active#(U191(X1, X2)) | → | U191#(active(X1), X2) | proper#(U181(X1, X2)) | → | U181#(proper(X1), proper(X2)) | |
| active#(U52(X1, X2)) | → | active#(X1) | proper#(U191(X1, X2)) | → | proper#(X2) | |
| active#(U201(tt, N, X, XS)) | → | splitAt#(N, XS) | active#(isPLNat(splitAt(V1, V2))) | → | U151#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2) | |
| active#(sel(X1, X2)) | → | active#(X1) | proper#(U202(X1, X2)) | → | proper#(X1) | |
| U52#(ok(X1), ok(X2)) | → | U52#(X1, X2) | proper#(U41(X1, X2, X3)) | → | proper#(X1) | |
| active#(splitAt(s(N), cons(X, XS))) | → | and#(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS))) | active#(U31(X1, X2)) | → | active#(X1) | |
| proper#(U132(X1, X2)) | → | proper#(X2) | take#(ok(X1), ok(X2)) | → | take#(X1, X2) | |
| U143#(mark(X)) | → | U143#(X) | U101#(ok(X1), ok(X2), ok(X3)) | → | U101#(X1, X2, X3) | |
| active#(sel(N, XS)) | → | isNatural#(N) | active#(sel(X1, X2)) | → | sel#(active(X1), X2) | |
| proper#(isPLNat(X)) | → | isPLNat#(proper(X)) | active#(U121(tt, V1)) | → | isNatural#(V1) | |
| snd#(mark(X)) | → | snd#(X) | proper#(U101(X1, X2, X3)) | → | proper#(X1) | |
| active#(head(cons(N, XS))) | → | isLNat#(XS) | proper#(isLNatKind(X)) | → | proper#(X) | |
| proper#(U112(X)) | → | proper#(X) | U141#(ok(X1), ok(X2), ok(X3)) | → | U141#(X1, X2, X3) | |
| active#(U131(X1, X2, X3)) | → | active#(X1) | proper#(U42(X1, X2)) | → | U42#(proper(X1), proper(X2)) | |
| active#(U181(X1, X2)) | → | U181#(active(X1), X2) | active#(tail(cons(N, XS))) | → | U211#(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS) | |
| proper#(U121(X1, X2)) | → | proper#(X2) | active#(U51(X1, X2, X3)) | → | U51#(active(X1), X2, X3) | |
| active#(afterNth(N, XS)) | → | isLNat#(XS) | proper#(U31(X1, X2)) | → | U31#(proper(X1), proper(X2)) | |
| proper#(U81(X1, X2)) | → | proper#(X1) | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) | |
| proper#(isPLNat(X)) | → | proper#(X) | proper#(U131(X1, X2, X3)) | → | U131#(proper(X1), proper(X2), proper(X3)) | |
| active#(isLNatKind(take(V1, V2))) | → | isLNatKind#(V2) | active#(U111(X1, X2)) | → | active#(X1) | |
| U211#(ok(X1), ok(X2)) | → | U211#(X1, X2) | active#(U142(tt, V2)) | → | isLNat#(V2) | |
| proper#(and(X1, X2)) | → | and#(proper(X1), proper(X2)) | proper#(U71(X1, X2)) | → | proper#(X2) | |
| active#(isNaturalKind(sel(V1, V2))) | → | isLNatKind#(V2) | active#(U191(X1, X2)) | → | active#(X1) | |
| active#(afterNth(N, XS)) | → | isNatural#(N) | proper#(and(X1, X2)) | → | proper#(X2) | |
| active#(isPLNat(pair(V1, V2))) | → | and#(isLNatKind(V1), isLNatKind(V2)) | U111#(ok(X1), ok(X2)) | → | U111#(X1, X2) | |
| proper#(U61(X1, X2)) | → | proper#(X2) | afterNth#(mark(X1), X2) | → | afterNth#(X1, X2) | |
| active#(isNatural(sel(V1, V2))) | → | isNaturalKind#(V1) | active#(U202(pair(YS, ZS), X)) | → | pair#(cons(X, YS), ZS) | |
| active#(pair(X1, X2)) | → | pair#(active(X1), X2) | U201#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U201#(X1, X2, X3, X4) | |
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | tail#(mark(X)) | → | tail#(X) | |
| proper#(pair(X1, X2)) | → | proper#(X1) | proper#(s(X)) | → | proper#(X) | |
| active#(snd(pair(X, Y))) | → | isLNatKind#(X) | active#(U201(tt, N, X, XS)) | → | U202#(splitAt(N, XS), X) | |
| active#(isNatural(head(V1))) | → | isLNatKind#(V1) | active#(U41(tt, V1, V2)) | → | isNatural#(V1) | |
| active#(snd(pair(X, Y))) | → | isLNat#(X) | proper#(U103(X)) | → | proper#(X) | |
| proper#(U221(X1, X2, X3)) | → | proper#(X1) | active#(U143(X)) | → | U143#(active(X)) | |
| U171#(ok(X1), ok(X2), ok(X3)) | → | U171#(X1, X2, X3) | proper#(natsFrom(X)) | → | natsFrom#(proper(X)) | |
| proper#(U72(X)) | → | U72#(proper(X)) | active#(splitAt(0, XS)) | → | and#(isLNat(XS), isLNatKind(XS)) | |
| U82#(ok(X)) | → | U82#(X) | active#(isNaturalKind(sel(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | |
| take#(mark(X1), X2) | → | take#(X1, X2) | proper#(U142(X1, X2)) | → | proper#(X2) | |
| U221#(mark(X1), X2, X3) | → | U221#(X1, X2, X3) | proper#(U151(X1, X2, X3)) | → | proper#(X1) | |
| proper#(U153(X)) | → | proper#(X) | active#(natsFrom(X)) | → | active#(X) | |
| active#(U102(X1, X2)) | → | U102#(active(X1), X2) | proper#(sel(X1, X2)) | → | sel#(proper(X1), proper(X2)) | |
| proper#(U152(X1, X2)) | → | proper#(X2) | proper#(U151(X1, X2, X3)) | → | proper#(X2) | |
| active#(U132(tt, V2)) | → | U133#(isLNat(V2)) | fst#(mark(X)) | → | fst#(X) | |
| active#(U171(tt, N, XS)) | → | afterNth#(N, XS) | U221#(ok(X1), ok(X2), ok(X3)) | → | U221#(X1, X2, X3) | |
| active#(U142(X1, X2)) | → | U142#(active(X1), X2) | proper#(U141(X1, X2, X3)) | → | proper#(X2) | |
| active#(U71(X1, X2)) | → | active#(X1) | U151#(mark(X1), X2, X3) | → | U151#(X1, X2, X3) | |
| active#(natsFrom(X)) | → | natsFrom#(active(X)) | U171#(mark(X1), X2, X3) | → | U171#(X1, X2, X3) | |
| active#(splitAt(s(N), cons(X, XS))) | → | isNatural#(X) | U121#(mark(X1), X2) | → | U121#(X1, X2) | |
| active#(fst(X)) | → | fst#(active(X)) | active#(isPLNat(splitAt(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | |
| proper#(U11(X1, X2, X3)) | → | U11#(proper(X1), proper(X2), proper(X3)) | active#(snd(pair(X, Y))) | → | and#(isLNat(Y), isLNatKind(Y)) | |
| active#(U131(tt, V1, V2)) | → | isNatural#(V1) | active#(isLNatKind(tail(V1))) | → | isLNatKind#(V1) | |
| active#(fst(pair(X, Y))) | → | isLNat#(X) | isPLNatKind#(ok(X)) | → | isPLNatKind#(X) | |
| proper#(U131(X1, X2, X3)) | → | proper#(X2) | active#(U201(X1, X2, X3, X4)) | → | U201#(active(X1), X2, X3, X4) | |
| proper#(afterNth(X1, X2)) | → | afterNth#(proper(X1), proper(X2)) | active#(afterNth(N, XS)) | → | isNaturalKind#(N) | |
| proper#(splitAt(X1, X2)) | → | proper#(X1) | U133#(ok(X)) | → | U133#(X) | |
| proper#(U112(X)) | → | U112#(proper(X)) | active#(tail(cons(N, XS))) | → | isNaturalKind#(N) | |
| proper#(U102(X1, X2)) | → | proper#(X2) | proper#(U211(X1, X2)) | → | proper#(X2) | |
| U161#(ok(X1), ok(X2)) | → | U161#(X1, X2) | active#(isNatural(s(V1))) | → | U121#(isNaturalKind(V1), V1) | |
| proper#(U221(X1, X2, X3)) | → | proper#(X2) | proper#(snd(X)) | → | snd#(proper(X)) | |
| active#(isPLNat(splitAt(V1, V2))) | → | isLNatKind#(V2) | proper#(U141(X1, X2, X3)) | → | U141#(proper(X1), proper(X2), proper(X3)) | |
| active#(sel(N, XS)) | → | and#(isNatural(N), isNaturalKind(N)) | active#(isNaturalKind(head(V1))) | → | isLNatKind#(V1) | |
| proper#(U122(X)) | → | U122#(proper(X)) | proper#(U143(X)) | → | proper#(X) | |
| proper#(U133(X)) | → | proper#(X) | active#(splitAt(0, XS)) | → | isLNatKind#(XS) | |
| active#(head(cons(N, XS))) | → | isNaturalKind#(N) | isNaturalKind#(ok(X)) | → | isNaturalKind#(X) | |
| proper#(U191(X1, X2)) | → | proper#(X1) | sel#(X1, mark(X2)) | → | sel#(X1, X2) | |
| U131#(mark(X1), X2, X3) | → | U131#(X1, X2, X3) | head#(ok(X)) | → | head#(X) | |
| active#(isNaturalKind(s(V1))) | → | isNaturalKind#(V1) | active#(U92(X)) | → | active#(X) | |
| U11#(ok(X1), ok(X2), ok(X3)) | → | U11#(X1, X2, X3) | proper#(U62(X)) | → | U62#(proper(X)) | |
| active#(U53(X)) | → | U53#(active(X)) | U143#(ok(X)) | → | U143#(X) | |
| proper#(U221(X1, X2, X3)) | → | U221#(proper(X1), proper(X2), proper(X3)) | pair#(mark(X1), X2) | → | pair#(X1, X2) | |
| U51#(ok(X1), ok(X2), ok(X3)) | → | U51#(X1, X2, X3) | U72#(ok(X)) | → | U72#(X) | |
| proper#(U201(X1, X2, X3, X4)) | → | proper#(X1) | cons#(mark(X1), X2) | → | cons#(X1, X2) | |
| active#(isLNatKind(natsFrom(V1))) | → | isNaturalKind#(V1) | active#(fst(pair(X, Y))) | → | isLNatKind#(Y) | |
| active#(U91(X1, X2)) | → | active#(X1) | active#(U132(X1, X2)) | → | U132#(active(X1), X2) | |
| U62#(mark(X)) | → | U62#(X) | natsFrom#(ok(X)) | → | natsFrom#(X) | |
| U71#(mark(X1), X2) | → | U71#(X1, X2) | proper#(U72(X)) | → | proper#(X) | |
| proper#(sel(X1, X2)) | → | proper#(X2) | proper#(U202(X1, X2)) | → | U202#(proper(X1), proper(X2)) | |
| active#(isLNat(afterNth(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | |
| proper#(U161(X1, X2)) | → | proper#(X1) | active#(U102(tt, V2)) | → | U103#(isLNat(V2)) | |
| head#(mark(X)) | → | head#(X) | isPLNat#(ok(X)) | → | isPLNat#(X) | |
| active#(s(X)) | → | s#(active(X)) | s#(ok(X)) | → | s#(X) | |
| proper#(sel(X1, X2)) | → | proper#(X1) | active#(take(X1, X2)) | → | take#(X1, active(X2)) | |
| active#(isNaturalKind(sel(V1, V2))) | → | isNaturalKind#(V1) | active#(splitAt(s(N), cons(X, XS))) | → | U201#(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS) | |
| active#(U72(X)) | → | active#(X) | U92#(mark(X)) | → | U92#(X) | |
| active#(natsFrom(N)) | → | U161#(and(isNatural(N), isNaturalKind(N)), N) | active#(U152(X1, X2)) | → | active#(X1) | |
| proper#(U31(X1, X2)) | → | proper#(X1) | proper#(natsFrom(X)) | → | proper#(X) | |
| active#(isLNatKind(cons(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | active#(U171(X1, X2, X3)) | → | U171#(active(X1), X2, X3) | |
| proper#(U161(X1, X2)) | → | U161#(proper(X1), proper(X2)) | U181#(ok(X1), ok(X2)) | → | U181#(X1, X2) | |
| active#(isLNatKind(take(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | active#(isPLNat(pair(V1, V2))) | → | isLNatKind#(V1) | |
| active#(U101(tt, V1, V2)) | → | isNatural#(V1) | active#(cons(X1, X2)) | → | cons#(active(X1), X2) | |
| active#(tail(X)) | → | tail#(active(X)) | active#(U111(X1, X2)) | → | U111#(active(X1), X2) | |
| active#(U61(X1, X2)) | → | U61#(active(X1), X2) | active#(U202(X1, X2)) | → | active#(X1) | |
| active#(U201(X1, X2, X3, X4)) | → | active#(X1) | active#(isLNat(snd(V1))) | → | U81#(isPLNatKind(V1), V1) | |
| U153#(mark(X)) | → | U153#(X) | active#(U62(X)) | → | U62#(active(X)) | |
| active#(isPLNat(splitAt(V1, V2))) | → | isNaturalKind#(V1) | top#(mark(X)) | → | top#(proper(X)) | |
| active#(U141(tt, V1, V2)) | → | U142#(isLNat(V1), V2) | active#(isLNat(cons(V1, V2))) | → | U51#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2) | |
| proper#(cons(X1, X2)) | → | proper#(X2) | active#(fst(pair(X, Y))) | → | and#(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))) | |
| active#(U153(X)) | → | U153#(active(X)) | U142#(mark(X1), X2) | → | U142#(X1, X2) | |
| active#(U151(tt, V1, V2)) | → | U152#(isNatural(V1), V2) | active#(U122(X)) | → | active#(X) | |
| proper#(afterNth(X1, X2)) | → | proper#(X2) | active#(tail(cons(N, XS))) | → | and#(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))) | |
| U72#(mark(X)) | → | U72#(X) | proper#(U101(X1, X2, X3)) | → | U101#(proper(X1), proper(X2), proper(X3)) | |
| proper#(snd(X)) | → | proper#(X) | proper#(head(X)) | → | head#(proper(X)) | |
| active#(isLNatKind(afterNth(V1, V2))) | → | and#(isNaturalKind(V1), isLNatKind(V2)) | proper#(splitAt(X1, X2)) | → | splitAt#(proper(X1), proper(X2)) | |
| proper#(U43(X)) | → | U43#(proper(X)) | proper#(U51(X1, X2, X3)) | → | U51#(proper(X1), proper(X2), proper(X3)) | |
| active#(snd(pair(X, Y))) | → | U181#(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y) | proper#(U211(X1, X2)) | → | U211#(proper(X1), proper(X2)) | |
| active#(U202(X1, X2)) | → | U202#(active(X1), X2) | U153#(ok(X)) | → | U153#(X) | |
| proper#(U82(X)) | → | U82#(proper(X)) | active#(U71(X1, X2)) | → | U71#(active(X1), X2) | |
| U161#(mark(X1), X2) | → | U161#(X1, X2) | proper#(U53(X)) | → | U53#(proper(X)) | |
| active#(isLNat(cons(V1, V2))) | → | isLNatKind#(V2) | active#(snd(pair(X, Y))) | → | and#(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))) | |
| active#(U42(X1, X2)) | → | active#(X1) | active#(U43(X)) | → | active#(X) | |
| U101#(mark(X1), X2, X3) | → | U101#(X1, X2, X3) | active#(U51(tt, V1, V2)) | → | isNatural#(V1) | |
| active#(afterNth(N, XS)) | → | and#(isLNat(XS), isLNatKind(XS)) | active#(afterNth(X1, X2)) | → | active#(X1) | |
| proper#(U111(X1, X2)) | → | proper#(X2) | active#(tail(cons(N, XS))) | → | isLNatKind#(XS) | |
| active#(isLNatKind(cons(V1, V2))) | → | isLNatKind#(V2) | U131#(ok(X1), ok(X2), ok(X3)) | → | U131#(X1, X2, X3) | |
| active#(U102(tt, V2)) | → | isLNat#(V2) | active#(fst(X)) | → | active#(X) | |
| active#(U152(tt, V2)) | → | U153#(isLNat(V2)) | active#(isLNat(take(V1, V2))) | → | U101#(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2) | |
| active#(U133(X)) | → | active#(X) | U31#(mark(X1), X2) | → | U31#(X1, X2) | |
| proper#(U61(X1, X2)) | → | proper#(X1) | fst#(ok(X)) | → | fst#(X) | |
| proper#(U141(X1, X2, X3)) | → | proper#(X1) | proper#(U132(X1, X2)) | → | U132#(proper(X1), proper(X2)) | |
| proper#(U52(X1, X2)) | → | proper#(X2) | active#(U121(X1, X2)) | → | U121#(active(X1), X2) | |
| active#(snd(pair(X, Y))) | → | and#(isLNat(X), isLNatKind(X)) | active#(U71(tt, V1)) | → | U72#(isNatural(V1)) | |
| active#(U21(X1, X2)) | → | active#(X1) | proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) | |
| active#(tail(cons(N, XS))) | → | and#(isNatural(N), isNaturalKind(N)) | proper#(U191(X1, X2)) | → | U191#(proper(X1), proper(X2)) | |
| proper#(U141(X1, X2, X3)) | → | proper#(X3) | active#(U11(tt, N, XS)) | → | snd#(splitAt(N, XS)) |
| active(U101(tt, V1, V2)) | → | mark(U102(isNatural(V1), V2)) | active(U102(tt, V2)) | → | mark(U103(isLNat(V2))) | |
| active(U103(tt)) | → | mark(tt) | active(U11(tt, N, XS)) | → | mark(snd(splitAt(N, XS))) | |
| active(U111(tt, V1)) | → | mark(U112(isLNat(V1))) | active(U112(tt)) | → | mark(tt) | |
| active(U121(tt, V1)) | → | mark(U122(isNatural(V1))) | active(U122(tt)) | → | mark(tt) | |
| active(U131(tt, V1, V2)) | → | mark(U132(isNatural(V1), V2)) | active(U132(tt, V2)) | → | mark(U133(isLNat(V2))) | |
| active(U133(tt)) | → | mark(tt) | active(U141(tt, V1, V2)) | → | mark(U142(isLNat(V1), V2)) | |
| active(U142(tt, V2)) | → | mark(U143(isLNat(V2))) | active(U143(tt)) | → | mark(tt) | |
| active(U151(tt, V1, V2)) | → | mark(U152(isNatural(V1), V2)) | active(U152(tt, V2)) | → | mark(U153(isLNat(V2))) | |
| active(U153(tt)) | → | mark(tt) | active(U161(tt, N)) | → | mark(cons(N, natsFrom(s(N)))) | |
| active(U171(tt, N, XS)) | → | mark(head(afterNth(N, XS))) | active(U181(tt, Y)) | → | mark(Y) | |
| active(U191(tt, XS)) | → | mark(pair(nil, XS)) | active(U201(tt, N, X, XS)) | → | mark(U202(splitAt(N, XS), X)) | |
| active(U202(pair(YS, ZS), X)) | → | mark(pair(cons(X, YS), ZS)) | active(U21(tt, X)) | → | mark(X) | |
| active(U211(tt, XS)) | → | mark(XS) | active(U221(tt, N, XS)) | → | mark(fst(splitAt(N, XS))) | |
| active(U31(tt, N)) | → | mark(N) | active(U41(tt, V1, V2)) | → | mark(U42(isNatural(V1), V2)) | |
| active(U42(tt, V2)) | → | mark(U43(isLNat(V2))) | active(U43(tt)) | → | mark(tt) | |
| active(U51(tt, V1, V2)) | → | mark(U52(isNatural(V1), V2)) | active(U52(tt, V2)) | → | mark(U53(isLNat(V2))) | |
| active(U53(tt)) | → | mark(tt) | active(U61(tt, V1)) | → | mark(U62(isPLNat(V1))) | |
| active(U62(tt)) | → | mark(tt) | active(U71(tt, V1)) | → | mark(U72(isNatural(V1))) | |
| active(U72(tt)) | → | mark(tt) | active(U81(tt, V1)) | → | mark(U82(isPLNat(V1))) | |
| active(U82(tt)) | → | mark(tt) | active(U91(tt, V1)) | → | mark(U92(isLNat(V1))) | |
| active(U92(tt)) | → | mark(tt) | active(afterNth(N, XS)) | → | mark(U11(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)) | |
| active(and(tt, X)) | → | mark(X) | active(fst(pair(X, Y))) | → | mark(U21(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), X)) | |
| active(head(cons(N, XS))) | → | mark(U31(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N)) | active(isLNat(nil)) | → | mark(tt) | |
| active(isLNat(afterNth(V1, V2))) | → | mark(U41(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)) | active(isLNat(cons(V1, V2))) | → | mark(U51(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)) | |
| active(isLNat(fst(V1))) | → | mark(U61(isPLNatKind(V1), V1)) | active(isLNat(natsFrom(V1))) | → | mark(U71(isNaturalKind(V1), V1)) | |
| active(isLNat(snd(V1))) | → | mark(U81(isPLNatKind(V1), V1)) | active(isLNat(tail(V1))) | → | mark(U91(isLNatKind(V1), V1)) | |
| active(isLNat(take(V1, V2))) | → | mark(U101(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)) | active(isLNatKind(nil)) | → | mark(tt) | |
| active(isLNatKind(afterNth(V1, V2))) | → | mark(and(isNaturalKind(V1), isLNatKind(V2))) | active(isLNatKind(cons(V1, V2))) | → | mark(and(isNaturalKind(V1), isLNatKind(V2))) | |
| active(isLNatKind(fst(V1))) | → | mark(isPLNatKind(V1)) | active(isLNatKind(natsFrom(V1))) | → | mark(isNaturalKind(V1)) | |
| active(isLNatKind(snd(V1))) | → | mark(isPLNatKind(V1)) | active(isLNatKind(tail(V1))) | → | mark(isLNatKind(V1)) | |
| active(isLNatKind(take(V1, V2))) | → | mark(and(isNaturalKind(V1), isLNatKind(V2))) | active(isNatural(0)) | → | mark(tt) | |
| active(isNatural(head(V1))) | → | mark(U111(isLNatKind(V1), V1)) | active(isNatural(s(V1))) | → | mark(U121(isNaturalKind(V1), V1)) | |
| active(isNatural(sel(V1, V2))) | → | mark(U131(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)) | active(isNaturalKind(0)) | → | mark(tt) | |
| active(isNaturalKind(head(V1))) | → | mark(isLNatKind(V1)) | active(isNaturalKind(s(V1))) | → | mark(isNaturalKind(V1)) | |
| active(isNaturalKind(sel(V1, V2))) | → | mark(and(isNaturalKind(V1), isLNatKind(V2))) | active(isPLNat(pair(V1, V2))) | → | mark(U141(and(isLNatKind(V1), isLNatKind(V2)), V1, V2)) | |
| active(isPLNat(splitAt(V1, V2))) | → | mark(U151(and(isNaturalKind(V1), isLNatKind(V2)), V1, V2)) | active(isPLNatKind(pair(V1, V2))) | → | mark(and(isLNatKind(V1), isLNatKind(V2))) | |
| active(isPLNatKind(splitAt(V1, V2))) | → | mark(and(isNaturalKind(V1), isLNatKind(V2))) | active(natsFrom(N)) | → | mark(U161(and(isNatural(N), isNaturalKind(N)), N)) | |
| active(sel(N, XS)) | → | mark(U171(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)) | active(snd(pair(X, Y))) | → | mark(U181(and(and(isLNat(X), isLNatKind(X)), and(isLNat(Y), isLNatKind(Y))), Y)) | |
| active(splitAt(0, XS)) | → | mark(U191(and(isLNat(XS), isLNatKind(XS)), XS)) | active(splitAt(s(N), cons(X, XS))) | → | mark(U201(and(and(isNatural(N), isNaturalKind(N)), and(and(isNatural(X), isNaturalKind(X)), and(isLNat(XS), isLNatKind(XS)))), N, X, XS)) | |
| active(tail(cons(N, XS))) | → | mark(U211(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), XS)) | active(take(N, XS)) | → | mark(U221(and(and(isNatural(N), isNaturalKind(N)), and(isLNat(XS), isLNatKind(XS))), N, XS)) | |
| active(U101(X1, X2, X3)) | → | U101(active(X1), X2, X3) | active(U102(X1, X2)) | → | U102(active(X1), X2) | |
| active(U103(X)) | → | U103(active(X)) | active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | |
| active(snd(X)) | → | snd(active(X)) | active(splitAt(X1, X2)) | → | splitAt(active(X1), X2) | |
| active(splitAt(X1, X2)) | → | splitAt(X1, active(X2)) | active(U111(X1, X2)) | → | U111(active(X1), X2) | |
| active(U112(X)) | → | U112(active(X)) | active(U121(X1, X2)) | → | U121(active(X1), X2) | |
| active(U122(X)) | → | U122(active(X)) | active(U131(X1, X2, X3)) | → | U131(active(X1), X2, X3) | |
| active(U132(X1, X2)) | → | U132(active(X1), X2) | active(U133(X)) | → | U133(active(X)) | |
| active(U141(X1, X2, X3)) | → | U141(active(X1), X2, X3) | active(U142(X1, X2)) | → | U142(active(X1), X2) | |
| active(U143(X)) | → | U143(active(X)) | active(U151(X1, X2, X3)) | → | U151(active(X1), X2, X3) | |
| active(U152(X1, X2)) | → | U152(active(X1), X2) | active(U153(X)) | → | U153(active(X)) | |
| active(U161(X1, X2)) | → | U161(active(X1), X2) | active(cons(X1, X2)) | → | cons(active(X1), X2) | |
| active(natsFrom(X)) | → | natsFrom(active(X)) | active(s(X)) | → | s(active(X)) | |
| active(U171(X1, X2, X3)) | → | U171(active(X1), X2, X3) | active(head(X)) | → | head(active(X)) | |
| active(afterNth(X1, X2)) | → | afterNth(active(X1), X2) | active(afterNth(X1, X2)) | → | afterNth(X1, active(X2)) | |
| active(U181(X1, X2)) | → | U181(active(X1), X2) | active(U191(X1, X2)) | → | U191(active(X1), X2) | |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | active(pair(X1, X2)) | → | pair(X1, active(X2)) | |
| active(U201(X1, X2, X3, X4)) | → | U201(active(X1), X2, X3, X4) | active(U202(X1, X2)) | → | U202(active(X1), X2) | |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | active(U211(X1, X2)) | → | U211(active(X1), X2) | |
| active(U221(X1, X2, X3)) | → | U221(active(X1), X2, X3) | active(fst(X)) | → | fst(active(X)) | |
| active(U31(X1, X2)) | → | U31(active(X1), X2) | 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)) | → | U61(active(X1), X2) | |
| active(U62(X)) | → | U62(active(X)) | active(U71(X1, X2)) | → | U71(active(X1), X2) | |
| active(U72(X)) | → | U72(active(X)) | active(U81(X1, X2)) | → | U81(active(X1), X2) | |
| active(U82(X)) | → | U82(active(X)) | active(U91(X1, X2)) | → | U91(active(X1), X2) | |
| active(U92(X)) | → | U92(active(X)) | active(and(X1, X2)) | → | and(active(X1), X2) | |
| active(tail(X)) | → | tail(active(X)) | active(take(X1, X2)) | → | take(active(X1), X2) | |
| active(take(X1, X2)) | → | take(X1, active(X2)) | active(sel(X1, X2)) | → | sel(active(X1), X2) | |
| active(sel(X1, X2)) | → | sel(X1, active(X2)) | U101(mark(X1), X2, X3) | → | mark(U101(X1, X2, X3)) | |
| U102(mark(X1), X2) | → | mark(U102(X1, X2)) | U103(mark(X)) | → | mark(U103(X)) | |
| U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) | snd(mark(X)) | → | mark(snd(X)) | |
| splitAt(mark(X1), X2) | → | mark(splitAt(X1, X2)) | splitAt(X1, mark(X2)) | → | mark(splitAt(X1, X2)) | |
| U111(mark(X1), X2) | → | mark(U111(X1, X2)) | U112(mark(X)) | → | mark(U112(X)) | |
| U121(mark(X1), X2) | → | mark(U121(X1, X2)) | U122(mark(X)) | → | mark(U122(X)) | |
| U131(mark(X1), X2, X3) | → | mark(U131(X1, X2, X3)) | U132(mark(X1), X2) | → | mark(U132(X1, X2)) | |
| U133(mark(X)) | → | mark(U133(X)) | U141(mark(X1), X2, X3) | → | mark(U141(X1, X2, X3)) | |
| U142(mark(X1), X2) | → | mark(U142(X1, X2)) | U143(mark(X)) | → | mark(U143(X)) | |
| U151(mark(X1), X2, X3) | → | mark(U151(X1, X2, X3)) | U152(mark(X1), X2) | → | mark(U152(X1, X2)) | |
| U153(mark(X)) | → | mark(U153(X)) | U161(mark(X1), X2) | → | mark(U161(X1, X2)) | |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | natsFrom(mark(X)) | → | mark(natsFrom(X)) | |
| s(mark(X)) | → | mark(s(X)) | U171(mark(X1), X2, X3) | → | mark(U171(X1, X2, X3)) | |
| head(mark(X)) | → | mark(head(X)) | afterNth(mark(X1), X2) | → | mark(afterNth(X1, X2)) | |
| afterNth(X1, mark(X2)) | → | mark(afterNth(X1, X2)) | U181(mark(X1), X2) | → | mark(U181(X1, X2)) | |
| U191(mark(X1), X2) | → | mark(U191(X1, X2)) | pair(mark(X1), X2) | → | mark(pair(X1, X2)) | |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | U201(mark(X1), X2, X3, X4) | → | mark(U201(X1, X2, X3, X4)) | |
| U202(mark(X1), X2) | → | mark(U202(X1, X2)) | U21(mark(X1), X2) | → | mark(U21(X1, X2)) | |
| U211(mark(X1), X2) | → | mark(U211(X1, X2)) | U221(mark(X1), X2, X3) | → | mark(U221(X1, X2, X3)) | |
| fst(mark(X)) | → | mark(fst(X)) | U31(mark(X1), X2) | → | mark(U31(X1, X2)) | |
| 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) | → | mark(U61(X1, X2)) | U62(mark(X)) | → | mark(U62(X)) | |
| U71(mark(X1), X2) | → | mark(U71(X1, X2)) | U72(mark(X)) | → | mark(U72(X)) | |
| U81(mark(X1), X2) | → | mark(U81(X1, X2)) | U82(mark(X)) | → | mark(U82(X)) | |
| U91(mark(X1), X2) | → | mark(U91(X1, X2)) | U92(mark(X)) | → | mark(U92(X)) | |
| and(mark(X1), X2) | → | mark(and(X1, X2)) | tail(mark(X)) | → | mark(tail(X)) | |
| take(mark(X1), X2) | → | mark(take(X1, X2)) | take(X1, mark(X2)) | → | mark(take(X1, X2)) | |
| sel(mark(X1), X2) | → | mark(sel(X1, X2)) | sel(X1, mark(X2)) | → | mark(sel(X1, X2)) | |
| proper(U101(X1, X2, X3)) | → | U101(proper(X1), proper(X2), proper(X3)) | proper(tt) | → | ok(tt) | |
| proper(U102(X1, X2)) | → | U102(proper(X1), proper(X2)) | proper(isNatural(X)) | → | isNatural(proper(X)) | |
| proper(U103(X)) | → | U103(proper(X)) | proper(isLNat(X)) | → | isLNat(proper(X)) | |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | proper(snd(X)) | → | snd(proper(X)) | |
| proper(splitAt(X1, X2)) | → | splitAt(proper(X1), proper(X2)) | proper(U111(X1, X2)) | → | U111(proper(X1), proper(X2)) | |
| proper(U112(X)) | → | U112(proper(X)) | proper(U121(X1, X2)) | → | U121(proper(X1), proper(X2)) | |
| proper(U122(X)) | → | U122(proper(X)) | proper(U131(X1, X2, X3)) | → | U131(proper(X1), proper(X2), proper(X3)) | |
| proper(U132(X1, X2)) | → | U132(proper(X1), proper(X2)) | proper(U133(X)) | → | U133(proper(X)) | |
| proper(U141(X1, X2, X3)) | → | U141(proper(X1), proper(X2), proper(X3)) | proper(U142(X1, X2)) | → | U142(proper(X1), proper(X2)) | |
| proper(U143(X)) | → | U143(proper(X)) | proper(U151(X1, X2, X3)) | → | U151(proper(X1), proper(X2), proper(X3)) | |
| proper(U152(X1, X2)) | → | U152(proper(X1), proper(X2)) | proper(U153(X)) | → | U153(proper(X)) | |
| proper(U161(X1, X2)) | → | U161(proper(X1), proper(X2)) | proper(cons(X1, X2)) | → | cons(proper(X1), proper(X2)) | |
| proper(natsFrom(X)) | → | natsFrom(proper(X)) | proper(s(X)) | → | s(proper(X)) | |
| proper(U171(X1, X2, X3)) | → | U171(proper(X1), proper(X2), proper(X3)) | proper(head(X)) | → | head(proper(X)) | |
| proper(afterNth(X1, X2)) | → | afterNth(proper(X1), proper(X2)) | proper(U181(X1, X2)) | → | U181(proper(X1), proper(X2)) | |
| proper(U191(X1, X2)) | → | U191(proper(X1), proper(X2)) | proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | |
| proper(nil) | → | ok(nil) | proper(U201(X1, X2, X3, X4)) | → | U201(proper(X1), proper(X2), proper(X3), proper(X4)) | |
| proper(U202(X1, X2)) | → | U202(proper(X1), proper(X2)) | proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | |
| proper(U211(X1, X2)) | → | U211(proper(X1), proper(X2)) | proper(U221(X1, X2, X3)) | → | U221(proper(X1), proper(X2), proper(X3)) | |
| proper(fst(X)) | → | fst(proper(X)) | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | |
| 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(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)) | → | U61(proper(X1), proper(X2)) | proper(U62(X)) | → | U62(proper(X)) | |
| proper(isPLNat(X)) | → | isPLNat(proper(X)) | proper(U71(X1, X2)) | → | U71(proper(X1), proper(X2)) | |
| proper(U72(X)) | → | U72(proper(X)) | proper(U81(X1, X2)) | → | U81(proper(X1), proper(X2)) | |
| proper(U82(X)) | → | U82(proper(X)) | proper(U91(X1, X2)) | → | U91(proper(X1), proper(X2)) | |
| proper(U92(X)) | → | U92(proper(X)) | proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) | |
| proper(isNaturalKind(X)) | → | isNaturalKind(proper(X)) | proper(isLNatKind(X)) | → | isLNatKind(proper(X)) | |
| proper(isPLNatKind(X)) | → | isPLNatKind(proper(X)) | proper(tail(X)) | → | tail(proper(X)) | |
| proper(take(X1, X2)) | → | take(proper(X1), proper(X2)) | proper(0) | → | ok(0) | |
| proper(sel(X1, X2)) | → | sel(proper(X1), proper(X2)) | U101(ok(X1), ok(X2), ok(X3)) | → | ok(U101(X1, X2, X3)) | |
| U102(ok(X1), ok(X2)) | → | ok(U102(X1, X2)) | isNatural(ok(X)) | → | ok(isNatural(X)) | |
| U103(ok(X)) | → | ok(U103(X)) | isLNat(ok(X)) | → | ok(isLNat(X)) | |
| U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | snd(ok(X)) | → | ok(snd(X)) | |
| splitAt(ok(X1), ok(X2)) | → | ok(splitAt(X1, X2)) | U111(ok(X1), ok(X2)) | → | ok(U111(X1, X2)) | |
| U112(ok(X)) | → | ok(U112(X)) | U121(ok(X1), ok(X2)) | → | ok(U121(X1, X2)) | |
| U122(ok(X)) | → | ok(U122(X)) | U131(ok(X1), ok(X2), ok(X3)) | → | ok(U131(X1, X2, X3)) | |
| U132(ok(X1), ok(X2)) | → | ok(U132(X1, X2)) | U133(ok(X)) | → | ok(U133(X)) | |
| U141(ok(X1), ok(X2), ok(X3)) | → | ok(U141(X1, X2, X3)) | U142(ok(X1), ok(X2)) | → | ok(U142(X1, X2)) | |
| U143(ok(X)) | → | ok(U143(X)) | U151(ok(X1), ok(X2), ok(X3)) | → | ok(U151(X1, X2, X3)) | |
| U152(ok(X1), ok(X2)) | → | ok(U152(X1, X2)) | U153(ok(X)) | → | ok(U153(X)) | |
| U161(ok(X1), ok(X2)) | → | ok(U161(X1, X2)) | cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | |
| natsFrom(ok(X)) | → | ok(natsFrom(X)) | s(ok(X)) | → | ok(s(X)) | |
| U171(ok(X1), ok(X2), ok(X3)) | → | ok(U171(X1, X2, X3)) | head(ok(X)) | → | ok(head(X)) | |
| afterNth(ok(X1), ok(X2)) | → | ok(afterNth(X1, X2)) | U181(ok(X1), ok(X2)) | → | ok(U181(X1, X2)) | |
| U191(ok(X1), ok(X2)) | → | ok(U191(X1, X2)) | pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | |
| U201(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U201(X1, X2, X3, X4)) | U202(ok(X1), ok(X2)) | → | ok(U202(X1, X2)) | |
| U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | U211(ok(X1), ok(X2)) | → | ok(U211(X1, X2)) | |
| U221(ok(X1), ok(X2), ok(X3)) | → | ok(U221(X1, X2, X3)) | fst(ok(X)) | → | ok(fst(X)) | |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | 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)) | |
| 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(U61(X1, X2)) | |
| U62(ok(X)) | → | ok(U62(X)) | isPLNat(ok(X)) | → | ok(isPLNat(X)) | |
| U71(ok(X1), ok(X2)) | → | ok(U71(X1, X2)) | U72(ok(X)) | → | ok(U72(X)) | |
| U81(ok(X1), ok(X2)) | → | ok(U81(X1, X2)) | U82(ok(X)) | → | ok(U82(X)) | |
| U91(ok(X1), ok(X2)) | → | ok(U91(X1, X2)) | U92(ok(X)) | → | ok(U92(X)) | |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | isNaturalKind(ok(X)) | → | ok(isNaturalKind(X)) | |
| isLNatKind(ok(X)) | → | ok(isLNatKind(X)) | isPLNatKind(ok(X)) | → | ok(isPLNatKind(X)) | |
| tail(ok(X)) | → | ok(tail(X)) | take(ok(X1), ok(X2)) | → | ok(take(X1, X2)) | |
| sel(ok(X1), ok(X2)) | → | ok(sel(X1, X2)) | top(mark(X)) | → | top(proper(X)) | |
| top(ok(X)) | → | top(active(X)) |
Termination of terms over the following signature is verified: natsFrom, U161, U112, fst, U111, U62, U61, U211, ok, proper, head, U21, U153, U151, mark, isLNatKind, isPLNat, U152, tail, and, U71, U191, U72, 0, isPLNatKind, U122, U121, U221, isLNat, active, U31, U181, pair, isNatural, U132, U131, U43, U42, U92, U133, U41, U91, top, snd, cons, isNaturalKind, U171, splitAt, U143, U201, U142, U202, U141, s, U51, tt, take, U82, U53, U81, U52, U11, afterNth, U102, sel, U103, U101, nil