Problem Transformed CSR 04 LISTUTILITIES complete FR

Tool CaT

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 LISTUTILITIES complete FR

stdout:

MAYBE

Problem:
U101(tt(),V1,V2) -> U102(isNatural(activate(V1)),activate(V2))
U102(tt(),V2) -> U103(isLNat(activate(V2)))
U103(tt()) -> tt()
U11(tt(),N,XS) -> snd(splitAt(activate(N),activate(XS)))
U111(tt(),V1) -> U112(isLNat(activate(V1)))
U112(tt()) -> tt()
U121(tt(),V1) -> U122(isNatural(activate(V1)))
U122(tt()) -> tt()
U131(tt(),V1,V2) -> U132(isNatural(activate(V1)),activate(V2))
U132(tt(),V2) -> U133(isLNat(activate(V2)))
U133(tt()) -> tt()
U141(tt(),V1,V2) -> U142(isLNat(activate(V1)),activate(V2))
U142(tt(),V2) -> U143(isLNat(activate(V2)))
U143(tt()) -> tt()
U151(tt(),V1,V2) -> U152(isNatural(activate(V1)),activate(V2))
U152(tt(),V2) -> U153(isLNat(activate(V2)))
U153(tt()) -> tt()
U161(tt(),N) -> cons(activate(N),n__natsFrom(n__s(activate(N))))
U171(tt(),N,XS) -> head(afterNth(activate(N),activate(XS)))
U181(tt(),Y) -> activate(Y)
U191(tt(),XS) -> pair(nil(),activate(XS))
U201(tt(),N,X,XS) -> U202(splitAt(activate(N),activate(XS)),activate(X))
U202(pair(YS,ZS),X) -> pair(cons(activate(X),YS),ZS)
U21(tt(),X) -> activate(X)
U211(tt(),XS) -> activate(XS)
U221(tt(),N,XS) -> fst(splitAt(activate(N),activate(XS)))
U31(tt(),N) -> activate(N)
U41(tt(),V1,V2) -> U42(isNatural(activate(V1)),activate(V2))
U42(tt(),V2) -> U43(isLNat(activate(V2)))
U43(tt()) -> tt()
U51(tt(),V1,V2) -> U52(isNatural(activate(V1)),activate(V2))
U52(tt(),V2) -> U53(isLNat(activate(V2)))
U53(tt()) -> tt()
U61(tt(),V1) -> U62(isPLNat(activate(V1)))
U62(tt()) -> tt()
U71(tt(),V1) -> U72(isNatural(activate(V1)))
U72(tt()) -> tt()
U81(tt(),V1) -> U82(isPLNat(activate(V1)))
U82(tt()) -> tt()
U91(tt(),V1) -> U92(isLNat(activate(V1)))
U92(tt()) -> tt()
afterNth(N,XS) ->
U11(and(and(isNatural(N),n__isNaturalKind(N)),n__and(n__isLNat(XS),n__isLNatKind(XS))),N,XS)
and(tt(),X) -> activate(X)
fst(pair(X,Y)) ->
U21(and(and(isLNat(X),n__isLNatKind(X)),n__and(n__isLNat(Y),n__isLNatKind(Y))),X)
head(cons(N,XS)) ->
U31(and(and(isNatural(N),n__isNaturalKind(N)),n__and(n__isLNat(activate(XS)),
                                                      n__isLNatKind(activate(XS)))),
     N)
isLNat(n__nil()) -> tt()
isLNat(n__afterNth(V1,V2)) ->
U41(and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2))),activate(V1),activate(V2))
isLNat(n__cons(V1,V2)) ->
U51(and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2))),activate(V1),activate(V2))
isLNat(n__fst(V1)) -> U61(isPLNatKind(activate(V1)),activate(V1))
isLNat(n__natsFrom(V1)) -> U71(isNaturalKind(activate(V1)),activate(V1))
isLNat(n__snd(V1)) -> U81(isPLNatKind(activate(V1)),activate(V1))
isLNat(n__tail(V1)) -> U91(isLNatKind(activate(V1)),activate(V1))
isLNat(n__take(V1,V2)) ->
U101(and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2))),activate(V1),activate(V2))
isLNatKind(n__nil()) -> tt()
isLNatKind(n__afterNth(V1,V2)) -> and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2)))
isLNatKind(n__cons(V1,V2)) -> and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2)))
isLNatKind(n__fst(V1)) -> isPLNatKind(activate(V1))
isLNatKind(n__natsFrom(V1)) -> isNaturalKind(activate(V1))
isLNatKind(n__snd(V1)) -> isPLNatKind(activate(V1))
isLNatKind(n__tail(V1)) -> isLNatKind(activate(V1))
isLNatKind(n__take(V1,V2)) -> and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2)))
isNatural(n__0()) -> tt()
isNatural(n__head(V1)) -> U111(isLNatKind(activate(V1)),activate(V1))
isNatural(n__s(V1)) -> U121(isNaturalKind(activate(V1)),activate(V1))
isNatural(n__sel(V1,V2)) ->
U131(and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2))),activate(V1),activate(V2))
isNaturalKind(n__0()) -> tt()
isNaturalKind(n__head(V1)) -> isLNatKind(activate(V1))
isNaturalKind(n__s(V1)) -> isNaturalKind(activate(V1))
isNaturalKind(n__sel(V1,V2)) -> and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2)))
isPLNat(n__pair(V1,V2)) ->
U141(and(isLNatKind(activate(V1)),n__isLNatKind(activate(V2))),activate(V1),activate(V2))
isPLNat(n__splitAt(V1,V2)) ->
U151(and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2))),activate(V1),activate(V2))
isPLNatKind(n__pair(V1,V2)) -> and(isLNatKind(activate(V1)),n__isLNatKind(activate(V2)))
isPLNatKind(n__splitAt(V1,V2)) -> and(isNaturalKind(activate(V1)),n__isLNatKind(activate(V2)))
natsFrom(N) -> U161(and(isNatural(N),n__isNaturalKind(N)),N)
sel(N,XS) ->
U171(and(and(isNatural(N),n__isNaturalKind(N)),n__and(n__isLNat(XS),n__isLNatKind(XS))),N,XS)
snd(pair(X,Y)) ->
U181(and(and(isLNat(X),n__isLNatKind(X)),n__and(n__isLNat(Y),n__isLNatKind(Y))),Y)
splitAt(0(),XS) -> U191(and(isLNat(XS),n__isLNatKind(XS)),XS)
splitAt(s(N),cons(X,XS)) ->
U201(and(and(isNatural(N),n__isNaturalKind(N)),n__and(n__and(n__isNatural(X),n__isNaturalKind(X)),
                                                       n__and(n__isLNat(activate(XS)),
                                                              n__isLNatKind
                                                              (activate(XS))))),
      N,X,activate(XS))
tail(cons(N,XS)) ->
U211(and(and(isNatural(N),n__isNaturalKind(N)),n__and(n__isLNat(activate(XS)),
                                                       n__isLNatKind(activate(XS)))),
      activate(XS))
take(N,XS) ->
U221(and(and(isNatural(N),n__isNaturalKind(N)),n__and(n__isLNat(XS),n__isLNatKind(XS))),N,XS)
natsFrom(X) -> n__natsFrom(X)
s(X) -> n__s(X)
isNaturalKind(X) -> n__isNaturalKind(X)
and(X1,X2) -> n__and(X1,X2)
isLNat(X) -> n__isLNat(X)
isLNatKind(X) -> n__isLNatKind(X)
nil() -> n__nil()
afterNth(X1,X2) -> n__afterNth(X1,X2)
cons(X1,X2) -> n__cons(X1,X2)
fst(X) -> n__fst(X)
snd(X) -> n__snd(X)
tail(X) -> n__tail(X)
take(X1,X2) -> n__take(X1,X2)
0() -> n__0()
head(X) -> n__head(X)
sel(X1,X2) -> n__sel(X1,X2)
pair(X1,X2) -> n__pair(X1,X2)
splitAt(X1,X2) -> n__splitAt(X1,X2)
isNatural(X) -> n__isNatural(X)
activate(n__natsFrom(X)) -> natsFrom(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__isNaturalKind(X)) -> isNaturalKind(X)
activate(n__and(X1,X2)) -> and(activate(X1),X2)
activate(n__isLNat(X)) -> isLNat(X)
activate(n__isLNatKind(X)) -> isLNatKind(X)
activate(n__nil()) -> nil()
activate(n__afterNth(X1,X2)) -> afterNth(activate(X1),activate(X2))
activate(n__cons(X1,X2)) -> cons(activate(X1),X2)
activate(n__fst(X)) -> fst(activate(X))
activate(n__snd(X)) -> snd(activate(X))
activate(n__tail(X)) -> tail(activate(X))
activate(n__take(X1,X2)) -> take(activate(X1),activate(X2))
activate(n__0()) -> 0()
activate(n__head(X)) -> head(activate(X))
activate(n__sel(X1,X2)) -> sel(activate(X1),activate(X2))
activate(n__pair(X1,X2)) -> pair(activate(X1),activate(X2))
activate(n__splitAt(X1,X2)) -> splitAt(activate(X1),activate(X2))
activate(n__isNatural(X)) -> isNatural(X)
activate(X) -> X

Proof:
Open

Tool IRC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 LISTUTILITIES complete FR

stdout:

MAYBE

Tool IRC2

Execution Time	Unknown
Answer	TIMEOUT
Input	Transformed CSR 04 LISTUTILITIES complete FR

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    innermost runtime-complexity with respect to
  Rules:
    {  U101(tt(), V1, V2) ->
       U102(isNatural(activate(V1)), activate(V2))
     , U102(tt(), V2) -> U103(isLNat(activate(V2)))
     , U103(tt()) -> tt()
     , U11(tt(), N, XS) -> snd(splitAt(activate(N), activate(XS)))
     , U111(tt(), V1) -> U112(isLNat(activate(V1)))
     , U112(tt()) -> tt()
     , U121(tt(), V1) -> U122(isNatural(activate(V1)))
     , U122(tt()) -> tt()
     , U131(tt(), V1, V2) -> U132(isNatural(activate(V1)), activate(V2))
     , U132(tt(), V2) -> U133(isLNat(activate(V2)))
     , U133(tt()) -> tt()
     , U141(tt(), V1, V2) -> U142(isLNat(activate(V1)), activate(V2))
     , U142(tt(), V2) -> U143(isLNat(activate(V2)))
     , U143(tt()) -> tt()
     , U151(tt(), V1, V2) -> U152(isNatural(activate(V1)), activate(V2))
     , U152(tt(), V2) -> U153(isLNat(activate(V2)))
     , U153(tt()) -> tt()
     , U161(tt(), N) ->
       cons(activate(N), n__natsFrom(n__s(activate(N))))
     , U171(tt(), N, XS) -> head(afterNth(activate(N), activate(XS)))
     , U181(tt(), Y) -> activate(Y)
     , U191(tt(), XS) -> pair(nil(), activate(XS))
     , U201(tt(), N, X, XS) ->
       U202(splitAt(activate(N), activate(XS)), activate(X))
     , U202(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
     , U21(tt(), X) -> activate(X)
     , U211(tt(), XS) -> activate(XS)
     , U221(tt(), N, XS) -> fst(splitAt(activate(N), activate(XS)))
     , U31(tt(), N) -> activate(N)
     , U41(tt(), V1, V2) -> U42(isNatural(activate(V1)), activate(V2))
     , U42(tt(), V2) -> U43(isLNat(activate(V2)))
     , U43(tt()) -> tt()
     , U51(tt(), V1, V2) -> U52(isNatural(activate(V1)), activate(V2))
     , U52(tt(), V2) -> U53(isLNat(activate(V2)))
     , U53(tt()) -> tt()
     , U61(tt(), V1) -> U62(isPLNat(activate(V1)))
     , U62(tt()) -> tt()
     , U71(tt(), V1) -> U72(isNatural(activate(V1)))
     , U72(tt()) -> tt()
     , U81(tt(), V1) -> U82(isPLNat(activate(V1)))
     , U82(tt()) -> tt()
     , U91(tt(), V1) -> U92(isLNat(activate(V1)))
     , U92(tt()) -> tt()
     , afterNth(N, XS) ->
       U11(and(and(isNatural(N), n__isNaturalKind(N)),
               n__and(n__isLNat(XS), n__isLNatKind(XS))),
           N,
           XS)
     , and(tt(), X) -> activate(X)
     , fst(pair(X, Y)) ->
       U21(and(and(isLNat(X), n__isLNatKind(X)),
               n__and(n__isLNat(Y), n__isLNatKind(Y))),
           X)
     , head(cons(N, XS)) ->
       U31(and(and(isNatural(N), n__isNaturalKind(N)),
               n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))),
           N)
     , isLNat(n__nil()) -> tt()
     , isLNat(n__afterNth(V1, V2)) ->
       U41(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , isLNat(n__cons(V1, V2)) ->
       U51(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , isLNat(n__fst(V1)) ->
       U61(isPLNatKind(activate(V1)), activate(V1))
     , isLNat(n__natsFrom(V1)) ->
       U71(isNaturalKind(activate(V1)), activate(V1))
     , isLNat(n__snd(V1)) ->
       U81(isPLNatKind(activate(V1)), activate(V1))
     , isLNat(n__tail(V1)) ->
       U91(isLNatKind(activate(V1)), activate(V1))
     , isLNat(n__take(V1, V2)) ->
       U101(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
            activate(V1),
            activate(V2))
     , isLNatKind(n__nil()) -> tt()
     , isLNatKind(n__afterNth(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isLNatKind(n__cons(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isLNatKind(n__fst(V1)) -> isPLNatKind(activate(V1))
     , isLNatKind(n__natsFrom(V1)) -> isNaturalKind(activate(V1))
     , isLNatKind(n__snd(V1)) -> isPLNatKind(activate(V1))
     , isLNatKind(n__tail(V1)) -> isLNatKind(activate(V1))
     , isLNatKind(n__take(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isNatural(n__0()) -> tt()
     , isNatural(n__head(V1)) ->
       U111(isLNatKind(activate(V1)), activate(V1))
     , isNatural(n__s(V1)) ->
       U121(isNaturalKind(activate(V1)), activate(V1))
     , isNatural(n__sel(V1, V2)) ->
       U131(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
            activate(V1),
            activate(V2))
     , isNaturalKind(n__0()) -> tt()
     , isNaturalKind(n__head(V1)) -> isLNatKind(activate(V1))
     , isNaturalKind(n__s(V1)) -> isNaturalKind(activate(V1))
     , isNaturalKind(n__sel(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isPLNat(n__pair(V1, V2)) ->
       U141(and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))),
            activate(V1),
            activate(V2))
     , isPLNat(n__splitAt(V1, V2)) ->
       U151(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
            activate(V1),
            activate(V2))
     , isPLNatKind(n__pair(V1, V2)) ->
       and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isPLNatKind(n__splitAt(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , natsFrom(N) -> U161(and(isNatural(N), n__isNaturalKind(N)), N)
     , sel(N, XS) ->
       U171(and(and(isNatural(N), n__isNaturalKind(N)),
                n__and(n__isLNat(XS), n__isLNatKind(XS))),
            N,
            XS)
     , snd(pair(X, Y)) ->
       U181(and(and(isLNat(X), n__isLNatKind(X)),
                n__and(n__isLNat(Y), n__isLNatKind(Y))),
            Y)
     , splitAt(0(), XS) -> U191(and(isLNat(XS), n__isLNatKind(XS)), XS)
     , splitAt(s(N), cons(X, XS)) ->
       U201(and(and(isNatural(N), n__isNaturalKind(N)),
                n__and(n__and(n__isNatural(X), n__isNaturalKind(X)),
                       n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS))))),
            N,
            X,
            activate(XS))
     , tail(cons(N, XS)) ->
       U211(and(and(isNatural(N), n__isNaturalKind(N)),
                n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))),
            activate(XS))
     , take(N, XS) ->
       U221(and(and(isNatural(N), n__isNaturalKind(N)),
                n__and(n__isLNat(XS), n__isLNatKind(XS))),
            N,
            XS)
     , natsFrom(X) -> n__natsFrom(X)
     , s(X) -> n__s(X)
     , isNaturalKind(X) -> n__isNaturalKind(X)
     , and(X1, X2) -> n__and(X1, X2)
     , isLNat(X) -> n__isLNat(X)
     , isLNatKind(X) -> n__isLNatKind(X)
     , nil() -> n__nil()
     , afterNth(X1, X2) -> n__afterNth(X1, X2)
     , cons(X1, X2) -> n__cons(X1, X2)
     , fst(X) -> n__fst(X)
     , snd(X) -> n__snd(X)
     , tail(X) -> n__tail(X)
     , take(X1, X2) -> n__take(X1, X2)
     , 0() -> n__0()
     , head(X) -> n__head(X)
     , sel(X1, X2) -> n__sel(X1, X2)
     , pair(X1, X2) -> n__pair(X1, X2)
     , splitAt(X1, X2) -> n__splitAt(X1, X2)
     , isNatural(X) -> n__isNatural(X)
     , activate(n__natsFrom(X)) -> natsFrom(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(n__isNaturalKind(X)) -> isNaturalKind(X)
     , activate(n__and(X1, X2)) -> and(activate(X1), X2)
     , activate(n__isLNat(X)) -> isLNat(X)
     , activate(n__isLNatKind(X)) -> isLNatKind(X)
     , activate(n__nil()) -> nil()
     , activate(n__afterNth(X1, X2)) ->
       afterNth(activate(X1), activate(X2))
     , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
     , activate(n__fst(X)) -> fst(activate(X))
     , activate(n__snd(X)) -> snd(activate(X))
     , activate(n__tail(X)) -> tail(activate(X))
     , activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
     , activate(n__0()) -> 0()
     , activate(n__head(X)) -> head(activate(X))
     , activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
     , activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2))
     , activate(n__splitAt(X1, X2)) ->
       splitAt(activate(X1), activate(X2))
     , activate(n__isNatural(X)) -> isNatural(X)
     , activate(X) -> X}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds

Tool RC1

Execution Time	Unknown
Answer	MAYBE
Input	Transformed CSR 04 LISTUTILITIES complete FR

stdout:

MAYBE

Tool RC2

Execution Time	Unknown
Answer	TIMEOUT
Input	Transformed CSR 04 LISTUTILITIES complete FR

stdout:

TIMEOUT

'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer:           TIMEOUT
Input Problem:    runtime-complexity with respect to
  Rules:
    {  U101(tt(), V1, V2) ->
       U102(isNatural(activate(V1)), activate(V2))
     , U102(tt(), V2) -> U103(isLNat(activate(V2)))
     , U103(tt()) -> tt()
     , U11(tt(), N, XS) -> snd(splitAt(activate(N), activate(XS)))
     , U111(tt(), V1) -> U112(isLNat(activate(V1)))
     , U112(tt()) -> tt()
     , U121(tt(), V1) -> U122(isNatural(activate(V1)))
     , U122(tt()) -> tt()
     , U131(tt(), V1, V2) -> U132(isNatural(activate(V1)), activate(V2))
     , U132(tt(), V2) -> U133(isLNat(activate(V2)))
     , U133(tt()) -> tt()
     , U141(tt(), V1, V2) -> U142(isLNat(activate(V1)), activate(V2))
     , U142(tt(), V2) -> U143(isLNat(activate(V2)))
     , U143(tt()) -> tt()
     , U151(tt(), V1, V2) -> U152(isNatural(activate(V1)), activate(V2))
     , U152(tt(), V2) -> U153(isLNat(activate(V2)))
     , U153(tt()) -> tt()
     , U161(tt(), N) ->
       cons(activate(N), n__natsFrom(n__s(activate(N))))
     , U171(tt(), N, XS) -> head(afterNth(activate(N), activate(XS)))
     , U181(tt(), Y) -> activate(Y)
     , U191(tt(), XS) -> pair(nil(), activate(XS))
     , U201(tt(), N, X, XS) ->
       U202(splitAt(activate(N), activate(XS)), activate(X))
     , U202(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
     , U21(tt(), X) -> activate(X)
     , U211(tt(), XS) -> activate(XS)
     , U221(tt(), N, XS) -> fst(splitAt(activate(N), activate(XS)))
     , U31(tt(), N) -> activate(N)
     , U41(tt(), V1, V2) -> U42(isNatural(activate(V1)), activate(V2))
     , U42(tt(), V2) -> U43(isLNat(activate(V2)))
     , U43(tt()) -> tt()
     , U51(tt(), V1, V2) -> U52(isNatural(activate(V1)), activate(V2))
     , U52(tt(), V2) -> U53(isLNat(activate(V2)))
     , U53(tt()) -> tt()
     , U61(tt(), V1) -> U62(isPLNat(activate(V1)))
     , U62(tt()) -> tt()
     , U71(tt(), V1) -> U72(isNatural(activate(V1)))
     , U72(tt()) -> tt()
     , U81(tt(), V1) -> U82(isPLNat(activate(V1)))
     , U82(tt()) -> tt()
     , U91(tt(), V1) -> U92(isLNat(activate(V1)))
     , U92(tt()) -> tt()
     , afterNth(N, XS) ->
       U11(and(and(isNatural(N), n__isNaturalKind(N)),
               n__and(n__isLNat(XS), n__isLNatKind(XS))),
           N,
           XS)
     , and(tt(), X) -> activate(X)
     , fst(pair(X, Y)) ->
       U21(and(and(isLNat(X), n__isLNatKind(X)),
               n__and(n__isLNat(Y), n__isLNatKind(Y))),
           X)
     , head(cons(N, XS)) ->
       U31(and(and(isNatural(N), n__isNaturalKind(N)),
               n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))),
           N)
     , isLNat(n__nil()) -> tt()
     , isLNat(n__afterNth(V1, V2)) ->
       U41(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , isLNat(n__cons(V1, V2)) ->
       U51(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
           activate(V1),
           activate(V2))
     , isLNat(n__fst(V1)) ->
       U61(isPLNatKind(activate(V1)), activate(V1))
     , isLNat(n__natsFrom(V1)) ->
       U71(isNaturalKind(activate(V1)), activate(V1))
     , isLNat(n__snd(V1)) ->
       U81(isPLNatKind(activate(V1)), activate(V1))
     , isLNat(n__tail(V1)) ->
       U91(isLNatKind(activate(V1)), activate(V1))
     , isLNat(n__take(V1, V2)) ->
       U101(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
            activate(V1),
            activate(V2))
     , isLNatKind(n__nil()) -> tt()
     , isLNatKind(n__afterNth(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isLNatKind(n__cons(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isLNatKind(n__fst(V1)) -> isPLNatKind(activate(V1))
     , isLNatKind(n__natsFrom(V1)) -> isNaturalKind(activate(V1))
     , isLNatKind(n__snd(V1)) -> isPLNatKind(activate(V1))
     , isLNatKind(n__tail(V1)) -> isLNatKind(activate(V1))
     , isLNatKind(n__take(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isNatural(n__0()) -> tt()
     , isNatural(n__head(V1)) ->
       U111(isLNatKind(activate(V1)), activate(V1))
     , isNatural(n__s(V1)) ->
       U121(isNaturalKind(activate(V1)), activate(V1))
     , isNatural(n__sel(V1, V2)) ->
       U131(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
            activate(V1),
            activate(V2))
     , isNaturalKind(n__0()) -> tt()
     , isNaturalKind(n__head(V1)) -> isLNatKind(activate(V1))
     , isNaturalKind(n__s(V1)) -> isNaturalKind(activate(V1))
     , isNaturalKind(n__sel(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isPLNat(n__pair(V1, V2)) ->
       U141(and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2))),
            activate(V1),
            activate(V2))
     , isPLNat(n__splitAt(V1, V2)) ->
       U151(and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2))),
            activate(V1),
            activate(V2))
     , isPLNatKind(n__pair(V1, V2)) ->
       and(isLNatKind(activate(V1)), n__isLNatKind(activate(V2)))
     , isPLNatKind(n__splitAt(V1, V2)) ->
       and(isNaturalKind(activate(V1)), n__isLNatKind(activate(V2)))
     , natsFrom(N) -> U161(and(isNatural(N), n__isNaturalKind(N)), N)
     , sel(N, XS) ->
       U171(and(and(isNatural(N), n__isNaturalKind(N)),
                n__and(n__isLNat(XS), n__isLNatKind(XS))),
            N,
            XS)
     , snd(pair(X, Y)) ->
       U181(and(and(isLNat(X), n__isLNatKind(X)),
                n__and(n__isLNat(Y), n__isLNatKind(Y))),
            Y)
     , splitAt(0(), XS) -> U191(and(isLNat(XS), n__isLNatKind(XS)), XS)
     , splitAt(s(N), cons(X, XS)) ->
       U201(and(and(isNatural(N), n__isNaturalKind(N)),
                n__and(n__and(n__isNatural(X), n__isNaturalKind(X)),
                       n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS))))),
            N,
            X,
            activate(XS))
     , tail(cons(N, XS)) ->
       U211(and(and(isNatural(N), n__isNaturalKind(N)),
                n__and(n__isLNat(activate(XS)), n__isLNatKind(activate(XS)))),
            activate(XS))
     , take(N, XS) ->
       U221(and(and(isNatural(N), n__isNaturalKind(N)),
                n__and(n__isLNat(XS), n__isLNatKind(XS))),
            N,
            XS)
     , natsFrom(X) -> n__natsFrom(X)
     , s(X) -> n__s(X)
     , isNaturalKind(X) -> n__isNaturalKind(X)
     , and(X1, X2) -> n__and(X1, X2)
     , isLNat(X) -> n__isLNat(X)
     , isLNatKind(X) -> n__isLNatKind(X)
     , nil() -> n__nil()
     , afterNth(X1, X2) -> n__afterNth(X1, X2)
     , cons(X1, X2) -> n__cons(X1, X2)
     , fst(X) -> n__fst(X)
     , snd(X) -> n__snd(X)
     , tail(X) -> n__tail(X)
     , take(X1, X2) -> n__take(X1, X2)
     , 0() -> n__0()
     , head(X) -> n__head(X)
     , sel(X1, X2) -> n__sel(X1, X2)
     , pair(X1, X2) -> n__pair(X1, X2)
     , splitAt(X1, X2) -> n__splitAt(X1, X2)
     , isNatural(X) -> n__isNatural(X)
     , activate(n__natsFrom(X)) -> natsFrom(activate(X))
     , activate(n__s(X)) -> s(activate(X))
     , activate(n__isNaturalKind(X)) -> isNaturalKind(X)
     , activate(n__and(X1, X2)) -> and(activate(X1), X2)
     , activate(n__isLNat(X)) -> isLNat(X)
     , activate(n__isLNatKind(X)) -> isLNatKind(X)
     , activate(n__nil()) -> nil()
     , activate(n__afterNth(X1, X2)) ->
       afterNth(activate(X1), activate(X2))
     , activate(n__cons(X1, X2)) -> cons(activate(X1), X2)
     , activate(n__fst(X)) -> fst(activate(X))
     , activate(n__snd(X)) -> snd(activate(X))
     , activate(n__tail(X)) -> tail(activate(X))
     , activate(n__take(X1, X2)) -> take(activate(X1), activate(X2))
     , activate(n__0()) -> 0()
     , activate(n__head(X)) -> head(activate(X))
     , activate(n__sel(X1, X2)) -> sel(activate(X1), activate(X2))
     , activate(n__pair(X1, X2)) -> pair(activate(X1), activate(X2))
     , activate(n__splitAt(X1, X2)) ->
       splitAt(activate(X1), activate(X2))
     , activate(n__isNatural(X)) -> isNatural(X)
     , activate(X) -> X}

Proof Output:
  Computation stopped due to timeout after 60.0 seconds