The TRS could not be proven terminating. The proof attempt took 60001 ms.
Problem 1 remains open; application of the following processors failed [DependencyGraph (timeout)].
| proper#(U41(X1, X2)) | → | proper#(X2) | proper#(U81(X1, X2, X3, X4)) | → | proper#(X3) | |
| proper#(cons(X1, X2)) | → | proper#(X1) | active#(tail(cons(N, XS))) | → | and#(isNatural(N), isLNat(XS)) | |
| proper#(U41(X1, X2)) | → | proper#(X1) | proper#(U11(X1, X2, X3)) | → | proper#(X3) | |
| proper#(tail(X)) | → | proper#(X) | proper#(U91(X1, X2)) | → | proper#(X2) | |
| active#(snd(pair(X, Y))) | → | and#(isLNat(X), isLNat(Y)) | tail#(ok(X)) | → | tail#(X) | |
| isNatural#(ok(X)) | → | isNatural#(X) | proper#(U11(X1, X2, X3)) | → | proper#(X2) | |
| active#(isPLNat(pair(V1, V2))) | → | isLNat#(V1) | top#(mark(X)) | → | proper#(X) | |
| active#(U11(tt, N, XS)) | → | splitAt#(N, XS) | active#(U101(X1, X2, X3)) | → | U101#(active(X1), X2, X3) | |
| proper#(U71(X1, X2)) | → | U71#(proper(X1), proper(X2)) | active#(U61(X1, X2)) | → | active#(X1) | |
| active#(tail(cons(N, XS))) | → | U91#(and(isNatural(N), isLNat(XS)), XS) | splitAt#(X1, mark(X2)) | → | splitAt#(X1, X2) | |
| and#(mark(X1), X2) | → | and#(X1, X2) | active#(afterNth(X1, X2)) | → | afterNth#(active(X1), X2) | |
| active#(U41(X1, X2)) | → | active#(X1) | active#(splitAt(0, XS)) | → | U71#(isLNat(XS), XS) | |
| sel#(ok(X1), ok(X2)) | → | sel#(X1, X2) | active#(isLNat(cons(V1, V2))) | → | isNatural#(V1) | |
| active#(take(N, XS)) | → | U101#(and(isNatural(N), isLNat(XS)), N, XS) | proper#(U101(X1, X2, X3)) | → | proper#(X2) | |
| proper#(splitAt(X1, X2)) | → | proper#(X2) | active#(U101(tt, N, XS)) | → | fst#(splitAt(N, XS)) | |
| active#(isLNat(afterNth(V1, V2))) | → | isNatural#(V1) | active#(isNatural(sel(V1, V2))) | → | isNatural#(V1) | |
| proper#(head(X)) | → | proper#(X) | natsFrom#(mark(X)) | → | natsFrom#(X) | |
| active#(U71(tt, XS)) | → | pair#(nil, XS) | active#(take(N, XS)) | → | isNatural#(N) | |
| active#(splitAt(0, XS)) | → | isLNat#(XS) | active#(U51(tt, N, XS)) | → | afterNth#(N, XS) | |
| active#(U81(X1, X2, X3, X4)) | → | U81#(active(X1), X2, X3, X4) | active#(and(X1, X2)) | → | and#(active(X1), X2) | |
| active#(snd(X)) | → | snd#(active(X)) | active#(splitAt(s(N), cons(X, XS))) | → | isLNat#(XS) | |
| active#(sel(X1, X2)) | → | active#(X1) | active#(isLNat(afterNth(V1, V2))) | → | isLNat#(V2) | |
| active#(U31(X1, X2)) | → | active#(X1) | take#(ok(X1), ok(X2)) | → | take#(X1, X2) | |
| U101#(ok(X1), ok(X2), ok(X3)) | → | U101#(X1, X2, X3) | active#(isNatural(sel(V1, V2))) | → | isLNat#(V2) | |
| isLNat#(ok(X)) | → | isLNat#(X) | active#(U21(X1, X2)) | → | U21#(active(X1), X2) | |
| active#(sel(N, XS)) | → | isNatural#(N) | active#(sel(X1, X2)) | → | sel#(active(X1), X2) | |
| active#(sel(X1, X2)) | → | sel#(X1, active(X2)) | proper#(isPLNat(X)) | → | isPLNat#(proper(X)) | |
| snd#(mark(X)) | → | snd#(X) | proper#(U101(X1, X2, X3)) | → | proper#(X1) | |
| active#(head(cons(N, XS))) | → | isLNat#(XS) | U91#(ok(X1), ok(X2)) | → | U91#(X1, X2) | |
| active#(afterNth(X1, X2)) | → | active#(X2) | proper#(pair(X1, X2)) | → | pair#(proper(X1), proper(X2)) | |
| proper#(tail(X)) | → | tail#(proper(X)) | active#(U51(X1, X2, X3)) | → | U51#(active(X1), X2, X3) | |
| proper#(U81(X1, X2, X3, X4)) | → | proper#(X4) | active#(afterNth(N, XS)) | → | isLNat#(XS) | |
| proper#(U31(X1, X2)) | → | U31#(proper(X1), proper(X2)) | active#(isLNat(take(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | |
| proper#(U21(X1, X2)) | → | proper#(X1) | U31#(ok(X1), ok(X2)) | → | U31#(X1, X2) | |
| proper#(isPLNat(X)) | → | proper#(X) | afterNth#(X1, mark(X2)) | → | afterNth#(X1, X2) | |
| active#(splitAt(s(N), cons(X, XS))) | → | and#(isNatural(N), and(isNatural(X), isLNat(XS))) | cons#(ok(X1), ok(X2)) | → | cons#(X1, X2) | |
| proper#(and(X1, X2)) | → | and#(proper(X1), proper(X2)) | U71#(ok(X1), ok(X2)) | → | U71#(X1, X2) | |
| proper#(U71(X1, X2)) | → | proper#(X2) | active#(head(cons(N, XS))) | → | isNatural#(N) | |
| proper#(isLNat(X)) | → | isLNat#(proper(X)) | active#(U41(X1, X2)) | → | U41#(active(X1), X2) | |
| active#(head(cons(N, XS))) | → | U31#(and(isNatural(N), isLNat(XS)), N) | active#(afterNth(N, XS)) | → | isNatural#(N) | |
| active#(sel(X1, X2)) | → | active#(X2) | proper#(and(X1, X2)) | → | proper#(X2) | |
| active#(isPLNat(pair(V1, V2))) | → | isLNat#(V2) | U91#(mark(X1), X2) | → | U91#(X1, X2) | |
| proper#(U61(X1, X2)) | → | proper#(X2) | U61#(mark(X1), X2) | → | U61#(X1, X2) | |
| active#(U101(X1, X2, X3)) | → | active#(X1) | afterNth#(mark(X1), X2) | → | afterNth#(X1, X2) | |
| proper#(afterNth(X1, X2)) | → | proper#(X1) | proper#(U82(X1, X2)) | → | proper#(X1) | |
| U61#(ok(X1), ok(X2)) | → | U61#(X1, X2) | active#(splitAt(s(N), cons(X, XS))) | → | isNatural#(N) | |
| active#(U51(X1, X2, X3)) | → | active#(X1) | proper#(fst(X)) | → | fst#(proper(X)) | |
| active#(isPLNat(pair(V1, V2))) | → | and#(isLNat(V1), isLNat(V2)) | proper#(pair(X1, X2)) | → | proper#(X2) | |
| active#(pair(X1, X2)) | → | pair#(active(X1), X2) | U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | |
| U81#(ok(X1), ok(X2), ok(X3), ok(X4)) | → | U81#(X1, X2, X3, X4) | active#(natsFrom(N)) | → | isNatural#(N) | |
| tail#(mark(X)) | → | tail#(X) | take#(X1, mark(X2)) | → | take#(X1, X2) | |
| proper#(pair(X1, X2)) | → | proper#(X1) | proper#(s(X)) | → | proper#(X) | |
| active#(snd(pair(X, Y))) | → | isLNat#(X) | proper#(U81(X1, X2, X3, X4)) | → | U81#(proper(X1), proper(X2), proper(X3), proper(X4)) | |
| proper#(natsFrom(X)) | → | natsFrom#(proper(X)) | proper#(take(X1, X2)) | → | take#(proper(X1), proper(X2)) | |
| active#(snd(X)) | → | active#(X) | active#(cons(X1, X2)) | → | active#(X1) | |
| sel#(mark(X1), X2) | → | sel#(X1, X2) | take#(mark(X1), X2) | → | take#(X1, X2) | |
| active#(take(N, XS)) | → | isLNat#(XS) | proper#(U31(X1, X2)) | → | proper#(X2) | |
| 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#(isLNat(fst(V1))) | → | isPLNat#(V1) | |
| active#(natsFrom(X)) | → | active#(X) | active#(isLNat(cons(V1, V2))) | → | isLNat#(V2) | |
| proper#(sel(X1, X2)) | → | sel#(proper(X1), proper(X2)) | splitAt#(ok(X1), ok(X2)) | → | splitAt#(X1, X2) | |
| U41#(ok(X1), ok(X2)) | → | U41#(X1, X2) | proper#(take(X1, X2)) | → | proper#(X1) | |
| fst#(mark(X)) | → | fst#(X) | active#(isPLNat(splitAt(V1, V2))) | → | isNatural#(V1) | |
| U41#(mark(X1), X2) | → | U41#(X1, X2) | active#(U51(tt, N, XS)) | → | head#(afterNth(N, XS)) | |
| proper#(U91(X1, X2)) | → | U91#(proper(X1), proper(X2)) | active#(U71(X1, X2)) | → | active#(X1) | |
| active#(natsFrom(X)) | → | natsFrom#(active(X)) | active#(splitAt(s(N), cons(X, XS))) | → | U81#(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS) | |
| active#(splitAt(s(N), cons(X, XS))) | → | isNatural#(X) | active#(fst(X)) | → | fst#(active(X)) | |
| proper#(take(X1, X2)) | → | proper#(X2) | active#(take(X1, X2)) | → | active#(X1) | |
| proper#(U41(X1, X2)) | → | U41#(proper(X1), proper(X2)) | proper#(U11(X1, X2, X3)) | → | U11#(proper(X1), proper(X2), proper(X3)) | |
| active#(U91(X1, X2)) | → | U91#(active(X1), X2) | active#(isLNat(cons(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | |
| 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#(U82(X1, X2)) | → | U82#(active(X1), X2) | proper#(U81(X1, X2, X3, X4)) | → | proper#(X1) | |
| active#(fst(pair(X, Y))) | → | isLNat#(X) | active#(tail(X)) | → | active#(X) | |
| active#(U82(X1, X2)) | → | active#(X1) | proper#(afterNth(X1, X2)) | → | afterNth#(proper(X1), proper(X2)) | |
| U21#(ok(X1), ok(X2)) | → | U21#(X1, X2) | proper#(splitAt(X1, X2)) | → | proper#(X1) | |
| active#(isLNat(take(V1, V2))) | → | isLNat#(V2) | proper#(isNatural(X)) | → | proper#(X) | |
| splitAt#(mark(X1), X2) | → | splitAt#(X1, X2) | proper#(snd(X)) | → | snd#(proper(X)) | |
| active#(take(X1, X2)) | → | take#(active(X1), X2) | active#(natsFrom(N)) | → | U41#(isNatural(N), N) | |
| active#(splitAt(X1, X2)) | → | splitAt#(active(X1), X2) | active#(splitAt(X1, X2)) | → | splitAt#(X1, active(X2)) | |
| active#(afterNth(N, XS)) | → | and#(isNatural(N), isLNat(XS)) | active#(isLNat(take(V1, V2))) | → | isNatural#(V1) | |
| active#(isLNat(natsFrom(V1))) | → | isNatural#(V1) | active#(afterNth(N, XS)) | → | U11#(and(isNatural(N), isLNat(XS)), N, XS) | |
| active#(U101(tt, N, XS)) | → | splitAt#(N, XS) | active#(take(X1, X2)) | → | active#(X2) | |
| active#(tail(cons(N, XS))) | → | isNatural#(N) | pair#(X1, mark(X2)) | → | pair#(X1, X2) | |
| sel#(X1, mark(X2)) | → | sel#(X1, X2) | head#(ok(X)) | → | head#(X) | |
| active#(isNatural(sel(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | proper#(U51(X1, X2, X3)) | → | proper#(X1) | |
| active#(U82(pair(YS, ZS), X)) | → | pair#(cons(X, YS), ZS) | U11#(ok(X1), ok(X2), ok(X3)) | → | U11#(X1, X2, X3) | |
| proper#(U101(X1, X2, X3)) | → | proper#(X3) | proper#(isNatural(X)) | → | isNatural#(proper(X)) | |
| proper#(U51(X1, X2, X3)) | → | proper#(X2) | pair#(mark(X1), X2) | → | pair#(X1, X2) | |
| U51#(ok(X1), ok(X2), ok(X3)) | → | U51#(X1, X2, X3) | cons#(mark(X1), X2) | → | cons#(X1, X2) | |
| active#(U81(tt, N, X, XS)) | → | splitAt#(N, XS) | active#(afterNth(X1, X2)) | → | afterNth#(X1, active(X2)) | |
| U82#(mark(X1), X2) | → | U82#(X1, X2) | top#(ok(X)) | → | active#(X) | |
| active#(U91(X1, X2)) | → | active#(X1) | proper#(U91(X1, X2)) | → | proper#(X1) | |
| active#(pair(X1, X2)) | → | active#(X1) | natsFrom#(ok(X)) | → | natsFrom#(X) | |
| U71#(mark(X1), X2) | → | U71#(X1, X2) | proper#(sel(X1, X2)) | → | proper#(X2) | |
| proper#(isLNat(X)) | → | proper#(X) | U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | |
| pair#(ok(X1), ok(X2)) | → | pair#(X1, X2) | head#(mark(X)) | → | head#(X) | |
| active#(head(X)) | → | head#(active(X)) | isPLNat#(ok(X)) | → | isPLNat#(X) | |
| active#(snd(pair(X, Y))) | → | isLNat#(Y) | active#(s(X)) | → | s#(active(X)) | |
| s#(ok(X)) | → | s#(X) | proper#(U11(X1, X2, X3)) | → | proper#(X1) | |
| proper#(sel(X1, X2)) | → | proper#(X1) | active#(take(X1, X2)) | → | take#(X1, active(X2)) | |
| proper#(s(X)) | → | s#(proper(X)) | active#(U81(X1, X2, X3, X4)) | → | active#(X1) | |
| active#(isNatural(s(V1))) | → | isNatural#(V1) | proper#(U81(X1, X2, X3, X4)) | → | proper#(X2) | |
| active#(splitAt(X1, X2)) | → | active#(X1) | U81#(mark(X1), X2, X3, X4) | → | U81#(X1, X2, X3, X4) | |
| proper#(U82(X1, X2)) | → | U82#(proper(X1), proper(X2)) | active#(isPLNat(splitAt(V1, V2))) | → | isLNat#(V2) | |
| top#(ok(X)) | → | top#(active(X)) | proper#(U31(X1, X2)) | → | proper#(X1) | |
| proper#(natsFrom(X)) | → | proper#(X) | active#(snd(pair(X, Y))) | → | U61#(and(isLNat(X), isLNat(Y)), Y) | |
| active#(cons(X1, X2)) | → | cons#(active(X1), X2) | active#(U41(tt, N)) | → | s#(N) | |
| active#(tail(X)) | → | tail#(active(X)) | active#(U61(X1, X2)) | → | U61#(active(X1), X2) | |
| active#(head(cons(N, XS))) | → | and#(isNatural(N), isLNat(XS)) | active#(pair(X1, X2)) | → | active#(X2) | |
| snd#(ok(X)) | → | snd#(X) | active#(head(X)) | → | active#(X) | |
| active#(U81(tt, N, X, XS)) | → | U82#(splitAt(N, XS), X) | active#(take(N, XS)) | → | and#(isNatural(N), isLNat(XS)) | |
| active#(U82(pair(YS, ZS), X)) | → | cons#(X, YS) | top#(mark(X)) | → | top#(proper(X)) | |
| active#(isNatural(head(V1))) | → | isLNat#(V1) | proper#(cons(X1, X2)) | → | proper#(X2) | |
| active#(isLNat(afterNth(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | proper#(afterNth(X1, X2)) | → | proper#(X2) | |
| active#(U41(tt, N)) | → | natsFrom#(s(N)) | active#(U11(X1, X2, X3)) | → | active#(X1) | |
| proper#(U101(X1, X2, X3)) | → | U101#(proper(X1), proper(X2), proper(X3)) | proper#(snd(X)) | → | proper#(X) | |
| proper#(head(X)) | → | head#(proper(X)) | proper#(splitAt(X1, X2)) | → | splitAt#(proper(X1), proper(X2)) | |
| active#(fst(pair(X, Y))) | → | and#(isLNat(X), isLNat(Y)) | proper#(U51(X1, X2, X3)) | → | U51#(proper(X1), proper(X2), proper(X3)) | |
| active#(sel(N, XS)) | → | U51#(and(isNatural(N), isLNat(XS)), N, XS) | proper#(U61(X1, X2)) | → | U61#(proper(X1), proper(X2)) | |
| active#(U71(X1, X2)) | → | U71#(active(X1), X2) | U82#(ok(X1), ok(X2)) | → | U82#(X1, X2) | |
| active#(splitAt(s(N), cons(X, XS))) | → | and#(isNatural(X), isLNat(XS)) | proper#(and(X1, X2)) | → | proper#(X1) | |
| U101#(mark(X1), X2, X3) | → | U101#(X1, X2, X3) | proper#(fst(X)) | → | proper#(X) | |
| afterNth#(ok(X1), ok(X2)) | → | afterNth#(X1, X2) | proper#(U21(X1, X2)) | → | proper#(X2) | |
| active#(isLNat(snd(V1))) | → | isPLNat#(V1) | active#(afterNth(X1, X2)) | → | active#(X1) | |
| active#(U31(X1, X2)) | → | U31#(active(X1), X2) | active#(splitAt(X1, X2)) | → | active#(X2) | |
| proper#(U71(X1, X2)) | → | proper#(X1) | proper#(U51(X1, X2, X3)) | → | proper#(X3) | |
| active#(isPLNat(splitAt(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | proper#(U82(X1, X2)) | → | proper#(X2) | |
| active#(sel(N, XS)) | → | isLNat#(XS) | proper#(U21(X1, X2)) | → | U21#(proper(X1), proper(X2)) | |
| active#(fst(pair(X, Y))) | → | U21#(and(isLNat(X), isLNat(Y)), X) | active#(U41(tt, N)) | → | cons#(N, natsFrom(s(N))) | |
| active#(fst(X)) | → | active#(X) | U31#(mark(X1), X2) | → | U31#(X1, X2) | |
| proper#(U61(X1, X2)) | → | proper#(X1) | fst#(ok(X)) | → | fst#(X) | |
| active#(U21(X1, X2)) | → | active#(X1) | proper#(cons(X1, X2)) | → | cons#(proper(X1), proper(X2)) | |
| active#(isLNat(tail(V1))) | → | isLNat#(V1) | U21#(mark(X1), X2) | → | U21#(X1, X2) | |
| active#(sel(N, XS)) | → | and#(isNatural(N), isLNat(XS)) | active#(U11(tt, N, XS)) | → | snd#(splitAt(N, XS)) | |
| active#(and(X1, X2)) | → | active#(X1) |
| active(U101(tt, N, XS)) | → | mark(fst(splitAt(N, XS))) | active(U11(tt, N, XS)) | → | mark(snd(splitAt(N, XS))) | |
| active(U21(tt, X)) | → | mark(X) | active(U31(tt, N)) | → | mark(N) | |
| active(U41(tt, N)) | → | mark(cons(N, natsFrom(s(N)))) | active(U51(tt, N, XS)) | → | mark(head(afterNth(N, XS))) | |
| active(U61(tt, Y)) | → | mark(Y) | active(U71(tt, XS)) | → | mark(pair(nil, XS)) | |
| active(U81(tt, N, X, XS)) | → | mark(U82(splitAt(N, XS), X)) | active(U82(pair(YS, ZS), X)) | → | mark(pair(cons(X, YS), ZS)) | |
| active(U91(tt, XS)) | → | mark(XS) | active(afterNth(N, XS)) | → | mark(U11(and(isNatural(N), isLNat(XS)), N, XS)) | |
| active(and(tt, X)) | → | mark(X) | active(fst(pair(X, Y))) | → | mark(U21(and(isLNat(X), isLNat(Y)), X)) | |
| active(head(cons(N, XS))) | → | mark(U31(and(isNatural(N), isLNat(XS)), N)) | active(isLNat(nil)) | → | mark(tt) | |
| active(isLNat(afterNth(V1, V2))) | → | mark(and(isNatural(V1), isLNat(V2))) | active(isLNat(cons(V1, V2))) | → | mark(and(isNatural(V1), isLNat(V2))) | |
| active(isLNat(fst(V1))) | → | mark(isPLNat(V1)) | active(isLNat(natsFrom(V1))) | → | mark(isNatural(V1)) | |
| active(isLNat(snd(V1))) | → | mark(isPLNat(V1)) | active(isLNat(tail(V1))) | → | mark(isLNat(V1)) | |
| active(isLNat(take(V1, V2))) | → | mark(and(isNatural(V1), isLNat(V2))) | active(isNatural(0)) | → | mark(tt) | |
| active(isNatural(head(V1))) | → | mark(isLNat(V1)) | active(isNatural(s(V1))) | → | mark(isNatural(V1)) | |
| active(isNatural(sel(V1, V2))) | → | mark(and(isNatural(V1), isLNat(V2))) | active(isPLNat(pair(V1, V2))) | → | mark(and(isLNat(V1), isLNat(V2))) | |
| active(isPLNat(splitAt(V1, V2))) | → | mark(and(isNatural(V1), isLNat(V2))) | active(natsFrom(N)) | → | mark(U41(isNatural(N), N)) | |
| active(sel(N, XS)) | → | mark(U51(and(isNatural(N), isLNat(XS)), N, XS)) | active(snd(pair(X, Y))) | → | mark(U61(and(isLNat(X), isLNat(Y)), Y)) | |
| active(splitAt(0, XS)) | → | mark(U71(isLNat(XS), XS)) | active(splitAt(s(N), cons(X, XS))) | → | mark(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) | |
| active(tail(cons(N, XS))) | → | mark(U91(and(isNatural(N), isLNat(XS)), XS)) | active(take(N, XS)) | → | mark(U101(and(isNatural(N), isLNat(XS)), N, XS)) | |
| active(U101(X1, X2, X3)) | → | U101(active(X1), X2, X3) | active(fst(X)) | → | fst(active(X)) | |
| active(splitAt(X1, X2)) | → | splitAt(active(X1), X2) | active(splitAt(X1, X2)) | → | splitAt(X1, active(X2)) | |
| active(U11(X1, X2, X3)) | → | U11(active(X1), X2, X3) | active(snd(X)) | → | snd(active(X)) | |
| active(U21(X1, X2)) | → | U21(active(X1), X2) | active(U31(X1, X2)) | → | U31(active(X1), X2) | |
| active(U41(X1, X2)) | → | U41(active(X1), X2) | active(cons(X1, X2)) | → | cons(active(X1), X2) | |
| active(natsFrom(X)) | → | natsFrom(active(X)) | active(s(X)) | → | s(active(X)) | |
| active(U51(X1, X2, X3)) | → | U51(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(U61(X1, X2)) | → | U61(active(X1), X2) | active(U71(X1, X2)) | → | U71(active(X1), X2) | |
| active(pair(X1, X2)) | → | pair(active(X1), X2) | active(pair(X1, X2)) | → | pair(X1, active(X2)) | |
| active(U81(X1, X2, X3, X4)) | → | U81(active(X1), X2, X3, X4) | active(U82(X1, X2)) | → | U82(active(X1), X2) | |
| active(U91(X1, X2)) | → | U91(active(X1), X2) | 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)) | |
| fst(mark(X)) | → | mark(fst(X)) | splitAt(mark(X1), X2) | → | mark(splitAt(X1, X2)) | |
| splitAt(X1, mark(X2)) | → | mark(splitAt(X1, X2)) | U11(mark(X1), X2, X3) | → | mark(U11(X1, X2, X3)) | |
| snd(mark(X)) | → | mark(snd(X)) | U21(mark(X1), X2) | → | mark(U21(X1, X2)) | |
| U31(mark(X1), X2) | → | mark(U31(X1, X2)) | U41(mark(X1), X2) | → | mark(U41(X1, X2)) | |
| cons(mark(X1), X2) | → | mark(cons(X1, X2)) | natsFrom(mark(X)) | → | mark(natsFrom(X)) | |
| s(mark(X)) | → | mark(s(X)) | U51(mark(X1), X2, X3) | → | mark(U51(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)) | U61(mark(X1), X2) | → | mark(U61(X1, X2)) | |
| U71(mark(X1), X2) | → | mark(U71(X1, X2)) | pair(mark(X1), X2) | → | mark(pair(X1, X2)) | |
| pair(X1, mark(X2)) | → | mark(pair(X1, X2)) | U81(mark(X1), X2, X3, X4) | → | mark(U81(X1, X2, X3, X4)) | |
| U82(mark(X1), X2) | → | mark(U82(X1, X2)) | U91(mark(X1), X2) | → | mark(U91(X1, X2)) | |
| 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(fst(X)) | → | fst(proper(X)) | proper(splitAt(X1, X2)) | → | splitAt(proper(X1), proper(X2)) | |
| proper(U11(X1, X2, X3)) | → | U11(proper(X1), proper(X2), proper(X3)) | proper(snd(X)) | → | snd(proper(X)) | |
| proper(U21(X1, X2)) | → | U21(proper(X1), proper(X2)) | proper(U31(X1, X2)) | → | U31(proper(X1), proper(X2)) | |
| proper(U41(X1, X2)) | → | U41(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(U51(X1, X2, X3)) | → | U51(proper(X1), proper(X2), proper(X3)) | proper(head(X)) | → | head(proper(X)) | |
| proper(afterNth(X1, X2)) | → | afterNth(proper(X1), proper(X2)) | proper(U61(X1, X2)) | → | U61(proper(X1), proper(X2)) | |
| proper(U71(X1, X2)) | → | U71(proper(X1), proper(X2)) | proper(pair(X1, X2)) | → | pair(proper(X1), proper(X2)) | |
| proper(nil) | → | ok(nil) | proper(U81(X1, X2, X3, X4)) | → | U81(proper(X1), proper(X2), proper(X3), proper(X4)) | |
| proper(U82(X1, X2)) | → | U82(proper(X1), proper(X2)) | proper(U91(X1, X2)) | → | U91(proper(X1), proper(X2)) | |
| proper(and(X1, X2)) | → | and(proper(X1), proper(X2)) | proper(isNatural(X)) | → | isNatural(proper(X)) | |
| proper(isLNat(X)) | → | isLNat(proper(X)) | proper(isPLNat(X)) | → | isPLNat(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)) | fst(ok(X)) | → | ok(fst(X)) | |
| splitAt(ok(X1), ok(X2)) | → | ok(splitAt(X1, X2)) | U11(ok(X1), ok(X2), ok(X3)) | → | ok(U11(X1, X2, X3)) | |
| snd(ok(X)) | → | ok(snd(X)) | U21(ok(X1), ok(X2)) | → | ok(U21(X1, X2)) | |
| U31(ok(X1), ok(X2)) | → | ok(U31(X1, X2)) | U41(ok(X1), ok(X2)) | → | ok(U41(X1, X2)) | |
| cons(ok(X1), ok(X2)) | → | ok(cons(X1, X2)) | natsFrom(ok(X)) | → | ok(natsFrom(X)) | |
| s(ok(X)) | → | ok(s(X)) | U51(ok(X1), ok(X2), ok(X3)) | → | ok(U51(X1, X2, X3)) | |
| head(ok(X)) | → | ok(head(X)) | afterNth(ok(X1), ok(X2)) | → | ok(afterNth(X1, X2)) | |
| U61(ok(X1), ok(X2)) | → | ok(U61(X1, X2)) | U71(ok(X1), ok(X2)) | → | ok(U71(X1, X2)) | |
| pair(ok(X1), ok(X2)) | → | ok(pair(X1, X2)) | U81(ok(X1), ok(X2), ok(X3), ok(X4)) | → | ok(U81(X1, X2, X3, X4)) | |
| U82(ok(X1), ok(X2)) | → | ok(U82(X1, X2)) | U91(ok(X1), ok(X2)) | → | ok(U91(X1, X2)) | |
| and(ok(X1), ok(X2)) | → | ok(and(X1, X2)) | isNatural(ok(X)) | → | ok(isNatural(X)) | |
| isLNat(ok(X)) | → | ok(isLNat(X)) | isPLNat(ok(X)) | → | ok(isPLNat(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: pair, natsFrom, isNatural, fst, U61, U41, U91, ok, proper, head, U21, top, snd, cons, mark, isPLNat, tail, splitAt, and, U71, 0, s, U51, tt, take, U82, isLNat, U81, U11, active, U31, afterNth, sel, U101, nil