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)].
| mark#(fst(X)) | → | active#(fst(mark(X))) | active#(tail(cons(N, XS))) | → | and#(isNatural(N), isLNat(XS)) | |
| U91#(X1, active(X2)) | → | U91#(X1, X2) | active#(isLNat(take(V1, V2))) | → | mark#(and(isNatural(V1), isLNat(V2))) | |
| active#(U82(pair(YS, ZS), X)) | → | mark#(pair(cons(X, YS), ZS)) | active#(snd(pair(X, Y))) | → | and#(isLNat(X), isLNat(Y)) | |
| mark#(U11(X1, X2, X3)) | → | U11#(mark(X1), X2, X3) | mark#(take(X1, X2)) | → | active#(take(mark(X1), mark(X2))) | |
| mark#(snd(X)) | → | active#(snd(mark(X))) | U21#(X1, active(X2)) | → | U21#(X1, X2) | |
| mark#(s(X)) | → | mark#(X) | mark#(sel(X1, X2)) | → | mark#(X1) | |
| active#(isPLNat(pair(V1, V2))) | → | isLNat#(V1) | active#(U11(tt, N, XS)) | → | splitAt#(N, XS) | |
| afterNth#(active(X1), X2) | → | afterNth#(X1, X2) | active#(fst(pair(X, Y))) | → | mark#(U21(and(isLNat(X), isLNat(Y)), X)) | |
| active#(tail(cons(N, XS))) | → | U91#(and(isNatural(N), isLNat(XS)), XS) | splitAt#(X1, mark(X2)) | → | splitAt#(X1, X2) | |
| active#(sel(N, XS)) | → | mark#(U51(and(isNatural(N), isLNat(XS)), N, XS)) | mark#(afterNth(X1, X2)) | → | mark#(X1) | |
| and#(mark(X1), X2) | → | and#(X1, X2) | U81#(X1, X2, X3, active(X4)) | → | U81#(X1, X2, X3, X4) | |
| active#(splitAt(0, XS)) | → | U71#(isLNat(XS), XS) | active#(isLNat(cons(V1, V2))) | → | mark#(and(isNatural(V1), isLNat(V2))) | |
| active#(isLNat(cons(V1, V2))) | → | isNatural#(V1) | active#(take(N, XS)) | → | U101#(and(isNatural(N), isLNat(XS)), N, XS) | |
| head#(active(X)) | → | head#(X) | U82#(X1, active(X2)) | → | U82#(X1, X2) | |
| active#(isNatural(sel(V1, V2))) | → | isNatural#(V1) | active#(U101(tt, N, XS)) | → | fst#(splitAt(N, XS)) | |
| active#(isLNat(afterNth(V1, V2))) | → | isNatural#(V1) | mark#(U11(X1, X2, X3)) | → | mark#(X1) | |
| natsFrom#(mark(X)) | → | natsFrom#(X) | active#(U71(tt, XS)) | → | pair#(nil, XS) | |
| isLNat#(active(X)) | → | isLNat#(X) | active#(take(N, XS)) | → | isNatural#(N) | |
| active#(splitAt(0, XS)) | → | isLNat#(XS) | active#(U51(tt, N, XS)) | → | afterNth#(N, XS) | |
| U91#(X1, mark(X2)) | → | U91#(X1, X2) | active#(isNatural(s(V1))) | → | mark#(isNatural(V1)) | |
| mark#(head(X)) | → | mark#(X) | mark#(splitAt(X1, X2)) | → | mark#(X1) | |
| mark#(U21(X1, X2)) | → | active#(U21(mark(X1), X2)) | active#(splitAt(s(N), cons(X, XS))) | → | isLNat#(XS) | |
| isNatural#(mark(X)) | → | isNatural#(X) | active#(isLNat(snd(V1))) | → | mark#(isPLNat(V1)) | |
| mark#(isNatural(X)) | → | isNatural#(X) | isPLNat#(active(X)) | → | isPLNat#(X) | |
| U11#(active(X1), X2, X3) | → | U11#(X1, X2, X3) | active#(isLNat(afterNth(V1, V2))) | → | isLNat#(V2) | |
| active#(U11(tt, N, XS)) | → | mark#(snd(splitAt(N, XS))) | cons#(X1, mark(X2)) | → | cons#(X1, X2) | |
| U41#(X1, active(X2)) | → | U41#(X1, X2) | mark#(U41(X1, X2)) | → | mark#(X1) | |
| U81#(active(X1), X2, X3, X4) | → | U81#(X1, X2, X3, X4) | active#(isNatural(sel(V1, V2))) | → | isLNat#(V2) | |
| mark#(fst(X)) | → | mark#(X) | active#(splitAt(0, XS)) | → | mark#(U71(isLNat(XS), XS)) | |
| active#(sel(N, XS)) | → | isNatural#(N) | mark#(U82(X1, X2)) | → | mark#(X1) | |
| snd#(mark(X)) | → | snd#(X) | mark#(U101(X1, X2, X3)) | → | U101#(mark(X1), X2, X3) | |
| active#(head(cons(N, XS))) | → | isLNat#(XS) | U21#(active(X1), X2) | → | U21#(X1, X2) | |
| U81#(X1, X2, X3, mark(X4)) | → | U81#(X1, X2, X3, X4) | pair#(X1, active(X2)) | → | pair#(X1, X2) | |
| mark#(snd(X)) | → | mark#(X) | cons#(active(X1), X2) | → | cons#(X1, X2) | |
| mark#(natsFrom(X)) | → | mark#(X) | mark#(afterNth(X1, X2)) | → | active#(afterNth(mark(X1), mark(X2))) | |
| active#(afterNth(N, XS)) | → | isLNat#(XS) | active#(isLNat(take(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | |
| mark#(take(X1, X2)) | → | mark#(X1) | U81#(X1, X2, mark(X3), X4) | → | U81#(X1, X2, X3, X4) | |
| mark#(U81(X1, X2, X3, X4)) | → | active#(U81(mark(X1), X2, X3, X4)) | afterNth#(X1, mark(X2)) | → | afterNth#(X1, X2) | |
| active#(splitAt(s(N), cons(X, XS))) | → | and#(isNatural(N), and(isNatural(X), isLNat(XS))) | isNatural#(active(X)) | → | isNatural#(X) | |
| U81#(X1, X2, active(X3), X4) | → | U81#(X1, X2, X3, X4) | U61#(active(X1), X2) | → | U61#(X1, X2) | |
| mark#(s(X)) | → | s#(mark(X)) | active#(head(cons(N, XS))) | → | isNatural#(N) | |
| mark#(U11(X1, X2, X3)) | → | active#(U11(mark(X1), X2, X3)) | active#(head(cons(N, XS))) | → | U31#(and(isNatural(N), isLNat(XS)), N) | |
| mark#(tail(X)) | → | active#(tail(mark(X))) | active#(afterNth(N, XS)) | → | isNatural#(N) | |
| active#(isPLNat(pair(V1, V2))) | → | isLNat#(V2) | U91#(mark(X1), X2) | → | U91#(X1, X2) | |
| active#(isLNat(fst(V1))) | → | mark#(isPLNat(V1)) | U61#(mark(X1), X2) | → | U61#(X1, X2) | |
| afterNth#(mark(X1), X2) | → | afterNth#(X1, X2) | mark#(natsFrom(X)) | → | active#(natsFrom(mark(X))) | |
| mark#(isLNat(X)) | → | active#(isLNat(X)) | U82#(X1, mark(X2)) | → | U82#(X1, X2) | |
| mark#(sel(X1, X2)) | → | sel#(mark(X1), mark(X2)) | active#(splitAt(s(N), cons(X, XS))) | → | isNatural#(N) | |
| U51#(X1, mark(X2), X3) | → | U51#(X1, X2, X3) | isPLNat#(mark(X)) | → | isPLNat#(X) | |
| U41#(X1, mark(X2)) | → | U41#(X1, X2) | mark#(isPLNat(X)) | → | isPLNat#(X) | |
| active#(isPLNat(pair(V1, V2))) | → | and#(isLNat(V1), isLNat(V2)) | mark#(and(X1, X2)) | → | and#(mark(X1), X2) | |
| U11#(mark(X1), X2, X3) | → | U11#(X1, X2, X3) | U101#(active(X1), X2, X3) | → | U101#(X1, X2, X3) | |
| tail#(mark(X)) | → | tail#(X) | active#(natsFrom(N)) | → | isNatural#(N) | |
| take#(X1, mark(X2)) | → | take#(X1, X2) | mark#(sel(X1, X2)) | → | active#(sel(mark(X1), mark(X2))) | |
| mark#(splitAt(X1, X2)) | → | active#(splitAt(mark(X1), mark(X2))) | mark#(U61(X1, X2)) | → | active#(U61(mark(X1), X2)) | |
| mark#(U71(X1, X2)) | → | mark#(X1) | cons#(X1, active(X2)) | → | cons#(X1, X2) | |
| active#(snd(pair(X, Y))) | → | isLNat#(X) | U51#(active(X1), X2, X3) | → | U51#(X1, X2, X3) | |
| U82#(active(X1), X2) | → | U82#(X1, X2) | mark#(U61(X1, X2)) | → | mark#(X1) | |
| sel#(mark(X1), X2) | → | sel#(X1, X2) | mark#(head(X)) | → | active#(head(mark(X))) | |
| take#(mark(X1), X2) | → | take#(X1, X2) | mark#(U101(X1, X2, X3)) | → | active#(U101(mark(X1), X2, X3)) | |
| and#(X1, active(X2)) | → | and#(X1, X2) | U101#(X1, active(X2), X3) | → | U101#(X1, X2, X3) | |
| mark#(pair(X1, X2)) | → | pair#(mark(X1), mark(X2)) | active#(take(N, XS)) | → | isLNat#(XS) | |
| mark#(U82(X1, X2)) | → | active#(U82(mark(X1), X2)) | mark#(U21(X1, X2)) | → | U21#(mark(X1), X2) | |
| mark#(U21(X1, X2)) | → | mark#(X1) | active#(isLNat(fst(V1))) | → | isPLNat#(V1) | |
| U61#(X1, mark(X2)) | → | U61#(X1, X2) | active#(isLNat(cons(V1, V2))) | → | isLNat#(V2) | |
| splitAt#(active(X1), X2) | → | splitAt#(X1, X2) | U11#(X1, X2, mark(X3)) | → | U11#(X1, X2, X3) | |
| mark#(nil) | → | active#(nil) | take#(X1, active(X2)) | → | take#(X1, X2) | |
| mark#(sel(X1, X2)) | → | mark#(X2) | fst#(mark(X)) | → | fst#(X) | |
| U101#(X1, X2, active(X3)) | → | U101#(X1, X2, X3) | active#(isPLNat(splitAt(V1, V2))) | → | isNatural#(V1) | |
| mark#(isNatural(X)) | → | active#(isNatural(X)) | mark#(U81(X1, X2, X3, X4)) | → | mark#(X1) | |
| afterNth#(X1, active(X2)) | → | afterNth#(X1, X2) | U41#(mark(X1), X2) | → | U41#(X1, X2) | |
| active#(U81(tt, N, X, XS)) | → | mark#(U82(splitAt(N, XS), X)) | active#(U61(tt, Y)) | → | mark#(Y) | |
| active#(U51(tt, N, XS)) | → | head#(afterNth(N, XS)) | mark#(cons(X1, X2)) | → | cons#(mark(X1), X2) | |
| and#(X1, mark(X2)) | → | and#(X1, X2) | natsFrom#(active(X)) | → | natsFrom#(X) | |
| active#(splitAt(s(N), cons(X, XS))) | → | U81#(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS) | active#(U41(tt, N)) | → | mark#(cons(N, natsFrom(s(N)))) | |
| active#(splitAt(s(N), cons(X, XS))) | → | isNatural#(X) | isLNat#(mark(X)) | → | isLNat#(X) | |
| mark#(isLNat(X)) | → | isLNat#(X) | active#(snd(pair(X, Y))) | → | mark#(U61(and(isLNat(X), isLNat(Y)), Y)) | |
| mark#(splitAt(X1, X2)) | → | mark#(X2) | active#(U101(tt, N, XS)) | → | mark#(fst(splitAt(N, XS))) | |
| mark#(head(X)) | → | head#(mark(X)) | mark#(take(X1, X2)) | → | take#(mark(X1), mark(X2)) | |
| s#(mark(X)) | → | s#(X) | active#(isLNat(cons(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | |
| active#(tail(cons(N, XS))) | → | mark#(U91(and(isNatural(N), isLNat(XS)), XS)) | active#(fst(pair(X, Y))) | → | isLNat#(Y) | |
| active#(tail(cons(N, XS))) | → | isLNat#(XS) | mark#(U31(X1, X2)) | → | U31#(mark(X1), X2) | |
| active#(fst(pair(X, Y))) | → | isLNat#(X) | mark#(U51(X1, X2, X3)) | → | U51#(mark(X1), X2, X3) | |
| active#(U31(tt, N)) | → | mark#(N) | active#(isLNat(afterNth(V1, V2))) | → | mark#(and(isNatural(V1), isLNat(V2))) | |
| mark#(U101(X1, X2, X3)) | → | mark#(X1) | mark#(U81(X1, X2, X3, X4)) | → | U81#(mark(X1), X2, X3, X4) | |
| active#(isLNat(take(V1, V2))) | → | isLNat#(V2) | mark#(pair(X1, X2)) | → | mark#(X2) | |
| U71#(X1, mark(X2)) | → | U71#(X1, X2) | splitAt#(mark(X1), X2) | → | splitAt#(X1, X2) | |
| active#(natsFrom(N)) | → | U41#(isNatural(N), N) | 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) | U51#(X1, X2, active(X3)) | → | U51#(X1, X2, X3) | |
| mark#(U71(X1, X2)) | → | U71#(mark(X1), X2) | U11#(X1, active(X2), X3) | → | U11#(X1, X2, X3) | |
| active#(U101(tt, N, XS)) | → | splitAt#(N, XS) | mark#(cons(X1, X2)) | → | mark#(X1) | |
| pair#(X1, mark(X2)) | → | pair#(X1, X2) | active#(tail(cons(N, XS))) | → | isNatural#(N) | |
| fst#(active(X)) | → | fst#(X) | U41#(active(X1), X2) | → | U41#(X1, X2) | |
| sel#(X1, mark(X2)) | → | sel#(X1, X2) | active#(isLNat(nil)) | → | mark#(tt) | |
| U81#(X1, mark(X2), X3, X4) | → | U81#(X1, X2, X3, X4) | mark#(and(X1, X2)) | → | mark#(X1) | |
| active#(isNatural(sel(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | mark#(pair(X1, X2)) | → | mark#(X1) | |
| sel#(X1, active(X2)) | → | sel#(X1, X2) | active#(U82(pair(YS, ZS), X)) | → | pair#(cons(X, YS), ZS) | |
| mark#(tail(X)) | → | mark#(X) | mark#(afterNth(X1, X2)) | → | mark#(X2) | |
| mark#(U82(X1, X2)) | → | U82#(mark(X1), X2) | pair#(mark(X1), X2) | → | pair#(X1, X2) | |
| cons#(mark(X1), X2) | → | cons#(X1, X2) | mark#(tt) | → | active#(tt) | |
| U31#(X1, active(X2)) | → | U31#(X1, X2) | active#(U81(tt, N, X, XS)) | → | splitAt#(N, XS) | |
| mark#(U61(X1, X2)) | → | U61#(mark(X1), X2) | mark#(tail(X)) | → | tail#(mark(X)) | |
| active#(isNatural(0)) | → | mark#(tt) | U82#(mark(X1), X2) | → | U82#(X1, X2) | |
| mark#(splitAt(X1, X2)) | → | splitAt#(mark(X1), mark(X2)) | U21#(X1, mark(X2)) | → | U21#(X1, X2) | |
| mark#(U51(X1, X2, X3)) | → | mark#(X1) | U71#(mark(X1), X2) | → | U71#(X1, X2) | |
| U61#(X1, active(X2)) | → | U61#(X1, X2) | U101#(X1, X2, mark(X3)) | → | U101#(X1, X2, X3) | |
| splitAt#(X1, active(X2)) | → | splitAt#(X1, X2) | mark#(U31(X1, X2)) | → | active#(U31(mark(X1), X2)) | |
| mark#(U91(X1, X2)) | → | U91#(mark(X1), X2) | U51#(mark(X1), X2, X3) | → | U51#(X1, X2, X3) | |
| mark#(0) | → | active#(0) | mark#(s(X)) | → | active#(s(mark(X))) | |
| head#(mark(X)) | → | head#(X) | active#(isNatural(sel(V1, V2))) | → | mark#(and(isNatural(V1), isLNat(V2))) | |
| active#(snd(pair(X, Y))) | → | isLNat#(Y) | active#(isNatural(head(V1))) | → | mark#(isLNat(V1)) | |
| active#(isPLNat(pair(V1, V2))) | → | mark#(and(isLNat(V1), isLNat(V2))) | snd#(active(X)) | → | snd#(X) | |
| active#(head(cons(N, XS))) | → | mark#(U31(and(isNatural(N), isLNat(XS)), N)) | mark#(U51(X1, X2, X3)) | → | active#(U51(mark(X1), X2, X3)) | |
| active#(isNatural(s(V1))) | → | isNatural#(V1) | U81#(mark(X1), X2, X3, X4) | → | U81#(X1, X2, X3, X4) | |
| U11#(X1, X2, active(X3)) | → | U11#(X1, X2, X3) | active#(isPLNat(splitAt(V1, V2))) | → | isLNat#(V2) | |
| mark#(cons(X1, X2)) | → | active#(cons(mark(X1), X2)) | mark#(fst(X)) | → | fst#(mark(X)) | |
| mark#(snd(X)) | → | snd#(mark(X)) | active#(snd(pair(X, Y))) | → | U61#(and(isLNat(X), isLNat(Y)), Y) | |
| mark#(U41(X1, X2)) | → | active#(U41(mark(X1), X2)) | U51#(X1, active(X2), X3) | → | U51#(X1, X2, X3) | |
| active#(isPLNat(splitAt(V1, V2))) | → | mark#(and(isNatural(V1), isLNat(V2))) | active#(U41(tt, N)) | → | s#(N) | |
| active#(head(cons(N, XS))) | → | and#(isNatural(N), isLNat(XS)) | U31#(active(X1), X2) | → | U31#(X1, X2) | |
| active#(natsFrom(N)) | → | mark#(U41(isNatural(N), N)) | mark#(U31(X1, X2)) | → | mark#(X1) | |
| active#(U81(tt, N, X, XS)) | → | U82#(splitAt(N, XS), X) | pair#(active(X1), X2) | → | pair#(X1, X2) | |
| mark#(U91(X1, X2)) | → | active#(U91(mark(X1), X2)) | active#(U82(pair(YS, ZS), X)) | → | cons#(X, YS) | |
| active#(take(N, XS)) | → | and#(isNatural(N), isLNat(XS)) | active#(U91(tt, XS)) | → | mark#(XS) | |
| active#(isNatural(head(V1))) | → | isLNat#(V1) | active#(take(N, XS)) | → | mark#(U101(and(isNatural(N), isLNat(XS)), N, XS)) | |
| active#(afterNth(N, XS)) | → | mark#(U11(and(isNatural(N), isLNat(XS)), N, XS)) | active#(isLNat(afterNth(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | |
| active#(U41(tt, N)) | → | natsFrom#(s(N)) | mark#(U41(X1, X2)) | → | U41#(mark(X1), X2) | |
| sel#(active(X1), X2) | → | sel#(X1, X2) | U71#(X1, active(X2)) | → | U71#(X1, X2) | |
| active#(fst(pair(X, Y))) | → | and#(isLNat(X), isLNat(Y)) | mark#(afterNth(X1, X2)) | → | afterNth#(mark(X1), mark(X2)) | |
| active#(splitAt(s(N), cons(X, XS))) | → | mark#(U81(and(isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS)) | active#(sel(N, XS)) | → | U51#(and(isNatural(N), isLNat(XS)), N, XS) | |
| active#(isLNat(tail(V1))) | → | mark#(isLNat(V1)) | and#(active(X1), X2) | → | and#(X1, X2) | |
| mark#(U91(X1, X2)) | → | mark#(X1) | active#(splitAt(s(N), cons(X, XS))) | → | and#(isNatural(X), isLNat(XS)) | |
| active#(U71(tt, XS)) | → | mark#(pair(nil, XS)) | U101#(mark(X1), X2, X3) | → | U101#(X1, X2, X3) | |
| U31#(X1, mark(X2)) | → | U31#(X1, X2) | active#(isLNat(snd(V1))) | → | isPLNat#(V1) | |
| U91#(active(X1), X2) | → | U91#(X1, X2) | active#(isLNat(natsFrom(V1))) | → | mark#(isNatural(V1)) | |
| tail#(active(X)) | → | tail#(X) | mark#(U71(X1, X2)) | → | active#(U71(mark(X1), X2)) | |
| mark#(and(X1, X2)) | → | active#(and(mark(X1), X2)) | active#(isPLNat(splitAt(V1, V2))) | → | and#(isNatural(V1), isLNat(V2)) | |
| active#(sel(N, XS)) | → | isLNat#(XS) | mark#(isPLNat(X)) | → | active#(isPLNat(X)) | |
| active#(U51(tt, N, XS)) | → | mark#(head(afterNth(N, XS))) | active#(U41(tt, N)) | → | cons#(N, natsFrom(s(N))) | |
| active#(fst(pair(X, Y))) | → | U21#(and(isLNat(X), isLNat(Y)), X) | U81#(X1, active(X2), X3, X4) | → | U81#(X1, X2, X3, X4) | |
| U101#(X1, mark(X2), X3) | → | U101#(X1, X2, X3) | U51#(X1, X2, mark(X3)) | → | U51#(X1, X2, X3) | |
| active#(and(tt, X)) | → | mark#(X) | U31#(mark(X1), X2) | → | U31#(X1, X2) | |
| active#(isLNat(tail(V1))) | → | isLNat#(V1) | U71#(active(X1), X2) | → | U71#(X1, X2) | |
| U21#(mark(X1), X2) | → | U21#(X1, X2) | active#(U21(tt, X)) | → | mark#(X) | |
| U11#(X1, mark(X2), X3) | → | U11#(X1, X2, X3) | mark#(take(X1, X2)) | → | mark#(X2) | |
| s#(active(X)) | → | s#(X) | active#(sel(N, XS)) | → | and#(isNatural(N), isLNat(XS)) | |
| active#(U11(tt, N, XS)) | → | snd#(splitAt(N, XS)) | mark#(natsFrom(X)) | → | natsFrom#(mark(X)) | |
| mark#(pair(X1, X2)) | → | active#(pair(mark(X1), mark(X2))) | take#(active(X1), X2) | → | take#(X1, X2) |
| 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)) | |
| mark(U101(X1, X2, X3)) | → | active(U101(mark(X1), X2, X3)) | mark(tt) | → | active(tt) | |
| mark(fst(X)) | → | active(fst(mark(X))) | mark(splitAt(X1, X2)) | → | active(splitAt(mark(X1), mark(X2))) | |
| mark(U11(X1, X2, X3)) | → | active(U11(mark(X1), X2, X3)) | mark(snd(X)) | → | active(snd(mark(X))) | |
| mark(U21(X1, X2)) | → | active(U21(mark(X1), X2)) | mark(U31(X1, X2)) | → | active(U31(mark(X1), X2)) | |
| mark(U41(X1, X2)) | → | active(U41(mark(X1), X2)) | mark(cons(X1, X2)) | → | active(cons(mark(X1), X2)) | |
| mark(natsFrom(X)) | → | active(natsFrom(mark(X))) | mark(s(X)) | → | active(s(mark(X))) | |
| mark(U51(X1, X2, X3)) | → | active(U51(mark(X1), X2, X3)) | mark(head(X)) | → | active(head(mark(X))) | |
| mark(afterNth(X1, X2)) | → | active(afterNth(mark(X1), mark(X2))) | mark(U61(X1, X2)) | → | active(U61(mark(X1), X2)) | |
| mark(U71(X1, X2)) | → | active(U71(mark(X1), X2)) | mark(pair(X1, X2)) | → | active(pair(mark(X1), mark(X2))) | |
| mark(nil) | → | active(nil) | mark(U81(X1, X2, X3, X4)) | → | active(U81(mark(X1), X2, X3, X4)) | |
| mark(U82(X1, X2)) | → | active(U82(mark(X1), X2)) | mark(U91(X1, X2)) | → | active(U91(mark(X1), X2)) | |
| mark(and(X1, X2)) | → | active(and(mark(X1), X2)) | mark(isNatural(X)) | → | active(isNatural(X)) | |
| mark(isLNat(X)) | → | active(isLNat(X)) | mark(isPLNat(X)) | → | active(isPLNat(X)) | |
| mark(tail(X)) | → | active(tail(mark(X))) | mark(take(X1, X2)) | → | active(take(mark(X1), mark(X2))) | |
| mark(0) | → | active(0) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
| U101(mark(X1), X2, X3) | → | U101(X1, X2, X3) | U101(X1, mark(X2), X3) | → | U101(X1, X2, X3) | |
| U101(X1, X2, mark(X3)) | → | U101(X1, X2, X3) | U101(active(X1), X2, X3) | → | U101(X1, X2, X3) | |
| U101(X1, active(X2), X3) | → | U101(X1, X2, X3) | U101(X1, X2, active(X3)) | → | U101(X1, X2, X3) | |
| fst(mark(X)) | → | fst(X) | fst(active(X)) | → | fst(X) | |
| splitAt(mark(X1), X2) | → | splitAt(X1, X2) | splitAt(X1, mark(X2)) | → | splitAt(X1, X2) | |
| splitAt(active(X1), X2) | → | splitAt(X1, X2) | splitAt(X1, active(X2)) | → | splitAt(X1, X2) | |
| U11(mark(X1), X2, X3) | → | U11(X1, X2, X3) | U11(X1, mark(X2), X3) | → | U11(X1, X2, X3) | |
| U11(X1, X2, mark(X3)) | → | U11(X1, X2, X3) | U11(active(X1), X2, X3) | → | U11(X1, X2, X3) | |
| U11(X1, active(X2), X3) | → | U11(X1, X2, X3) | U11(X1, X2, active(X3)) | → | U11(X1, X2, X3) | |
| snd(mark(X)) | → | snd(X) | snd(active(X)) | → | snd(X) | |
| U21(mark(X1), X2) | → | U21(X1, X2) | U21(X1, mark(X2)) | → | U21(X1, X2) | |
| U21(active(X1), X2) | → | U21(X1, X2) | U21(X1, active(X2)) | → | U21(X1, X2) | |
| U31(mark(X1), X2) | → | U31(X1, X2) | U31(X1, mark(X2)) | → | U31(X1, X2) | |
| U31(active(X1), X2) | → | U31(X1, X2) | U31(X1, active(X2)) | → | U31(X1, X2) | |
| U41(mark(X1), X2) | → | U41(X1, X2) | U41(X1, mark(X2)) | → | U41(X1, X2) | |
| U41(active(X1), X2) | → | U41(X1, X2) | U41(X1, active(X2)) | → | U41(X1, X2) | |
| cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
| cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
| natsFrom(mark(X)) | → | natsFrom(X) | natsFrom(active(X)) | → | natsFrom(X) | |
| s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
| U51(mark(X1), X2, X3) | → | U51(X1, X2, X3) | U51(X1, mark(X2), X3) | → | U51(X1, X2, X3) | |
| U51(X1, X2, mark(X3)) | → | U51(X1, X2, X3) | U51(active(X1), X2, X3) | → | U51(X1, X2, X3) | |
| U51(X1, active(X2), X3) | → | U51(X1, X2, X3) | U51(X1, X2, active(X3)) | → | U51(X1, X2, X3) | |
| head(mark(X)) | → | head(X) | head(active(X)) | → | head(X) | |
| afterNth(mark(X1), X2) | → | afterNth(X1, X2) | afterNth(X1, mark(X2)) | → | afterNth(X1, X2) | |
| afterNth(active(X1), X2) | → | afterNth(X1, X2) | afterNth(X1, active(X2)) | → | afterNth(X1, X2) | |
| U61(mark(X1), X2) | → | U61(X1, X2) | U61(X1, mark(X2)) | → | U61(X1, X2) | |
| U61(active(X1), X2) | → | U61(X1, X2) | U61(X1, active(X2)) | → | U61(X1, X2) | |
| U71(mark(X1), X2) | → | U71(X1, X2) | U71(X1, mark(X2)) | → | U71(X1, X2) | |
| U71(active(X1), X2) | → | U71(X1, X2) | U71(X1, active(X2)) | → | U71(X1, X2) | |
| pair(mark(X1), X2) | → | pair(X1, X2) | pair(X1, mark(X2)) | → | pair(X1, X2) | |
| pair(active(X1), X2) | → | pair(X1, X2) | pair(X1, active(X2)) | → | pair(X1, X2) | |
| U81(mark(X1), X2, X3, X4) | → | U81(X1, X2, X3, X4) | U81(X1, mark(X2), X3, X4) | → | U81(X1, X2, X3, X4) | |
| U81(X1, X2, mark(X3), X4) | → | U81(X1, X2, X3, X4) | U81(X1, X2, X3, mark(X4)) | → | U81(X1, X2, X3, X4) | |
| U81(active(X1), X2, X3, X4) | → | U81(X1, X2, X3, X4) | U81(X1, active(X2), X3, X4) | → | U81(X1, X2, X3, X4) | |
| U81(X1, X2, active(X3), X4) | → | U81(X1, X2, X3, X4) | U81(X1, X2, X3, active(X4)) | → | U81(X1, X2, X3, X4) | |
| U82(mark(X1), X2) | → | U82(X1, X2) | U82(X1, mark(X2)) | → | U82(X1, X2) | |
| U82(active(X1), X2) | → | U82(X1, X2) | U82(X1, active(X2)) | → | U82(X1, X2) | |
| U91(mark(X1), X2) | → | U91(X1, X2) | U91(X1, mark(X2)) | → | U91(X1, X2) | |
| U91(active(X1), X2) | → | U91(X1, X2) | U91(X1, active(X2)) | → | U91(X1, X2) | |
| and(mark(X1), X2) | → | and(X1, X2) | and(X1, mark(X2)) | → | and(X1, X2) | |
| and(active(X1), X2) | → | and(X1, X2) | and(X1, active(X2)) | → | and(X1, X2) | |
| isNatural(mark(X)) | → | isNatural(X) | isNatural(active(X)) | → | isNatural(X) | |
| isLNat(mark(X)) | → | isLNat(X) | isLNat(active(X)) | → | isLNat(X) | |
| isPLNat(mark(X)) | → | isPLNat(X) | isPLNat(active(X)) | → | isPLNat(X) | |
| tail(mark(X)) | → | tail(X) | tail(active(X)) | → | tail(X) | |
| take(mark(X1), X2) | → | take(X1, X2) | take(X1, mark(X2)) | → | take(X1, X2) | |
| take(active(X1), X2) | → | take(X1, X2) | take(X1, active(X2)) | → | take(X1, X2) | |
| sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
| sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) |
Termination of terms over the following signature is verified: pair, natsFrom, isNatural, fst, U61, U41, U91, head, U21, snd, cons, mark, isPLNat, tail, splitAt, and, U71, 0, s, U51, tt, take, U82, isLNat, U81, U11, active, U31, afterNth, sel, U101, nil