/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) QDP
(5) TransformationProof [EQUIVALENT, 0 ms]
(6) QDP
(7) TransformationProof [EQUIVALENT, 0 ms]
(8) QDP
(9) DependencyGraphProof [EQUIVALENT, 0 ms]
(10) QDP
(11) TransformationProof [EQUIVALENT, 0 ms]
(12) QDP
(13) TransformationProof [EQUIVALENT, 0 ms]
(14) QDP
(15) DependencyGraphProof [EQUIVALENT, 0 ms]
(16) QDP
(17) QDPOrderProof [EQUIVALENT, 35 ms]
(18) QDP
(19) DependencyGraphProof [EQUIVALENT, 0 ms]
(20) TRUE


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.

----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U11^1(tt, N, XS) -> U12^1(tt, activate(N), activate(XS))
   U11^1(tt, N, XS) -> ACTIVATE(N)
   U11^1(tt, N, XS) -> ACTIVATE(XS)
   U12^1(tt, N, XS) -> SND(splitAt(activate(N), activate(XS)))
   U12^1(tt, N, XS) -> SPLITAT(activate(N), activate(XS))
   U12^1(tt, N, XS) -> ACTIVATE(N)
   U12^1(tt, N, XS) -> ACTIVATE(XS)
   U21^1(tt, X) -> U22^1(tt, activate(X))
   U21^1(tt, X) -> ACTIVATE(X)
   U22^1(tt, X) -> ACTIVATE(X)
   U31^1(tt, N) -> U32^1(tt, activate(N))
   U31^1(tt, N) -> ACTIVATE(N)
   U32^1(tt, N) -> ACTIVATE(N)
   U41^1(tt, N, XS) -> U42^1(tt, activate(N), activate(XS))
   U41^1(tt, N, XS) -> ACTIVATE(N)
   U41^1(tt, N, XS) -> ACTIVATE(XS)
   U42^1(tt, N, XS) -> HEAD(afterNth(activate(N), activate(XS)))
   U42^1(tt, N, XS) -> AFTERNTH(activate(N), activate(XS))
   U42^1(tt, N, XS) -> ACTIVATE(N)
   U42^1(tt, N, XS) -> ACTIVATE(XS)
   U51^1(tt, Y) -> U52^1(tt, activate(Y))
   U51^1(tt, Y) -> ACTIVATE(Y)
   U52^1(tt, Y) -> ACTIVATE(Y)
   U61^1(tt, N, X, XS) -> U62^1(tt, activate(N), activate(X), activate(XS))
   U61^1(tt, N, X, XS) -> ACTIVATE(N)
   U61^1(tt, N, X, XS) -> ACTIVATE(X)
   U61^1(tt, N, X, XS) -> ACTIVATE(XS)
   U62^1(tt, N, X, XS) -> U63^1(tt, activate(N), activate(X), activate(XS))
   U62^1(tt, N, X, XS) -> ACTIVATE(N)
   U62^1(tt, N, X, XS) -> ACTIVATE(X)
   U62^1(tt, N, X, XS) -> ACTIVATE(XS)
   U63^1(tt, N, X, XS) -> U64^1(splitAt(activate(N), activate(XS)), activate(X))
   U63^1(tt, N, X, XS) -> SPLITAT(activate(N), activate(XS))
   U63^1(tt, N, X, XS) -> ACTIVATE(N)
   U63^1(tt, N, X, XS) -> ACTIVATE(XS)
   U63^1(tt, N, X, XS) -> ACTIVATE(X)
   U64^1(pair(YS, ZS), X) -> ACTIVATE(X)
   U71^1(tt, XS) -> U72^1(tt, activate(XS))
   U71^1(tt, XS) -> ACTIVATE(XS)
   U72^1(tt, XS) -> ACTIVATE(XS)
   U81^1(tt, N, XS) -> U82^1(tt, activate(N), activate(XS))
   U81^1(tt, N, XS) -> ACTIVATE(N)
   U81^1(tt, N, XS) -> ACTIVATE(XS)
   U82^1(tt, N, XS) -> FST(splitAt(activate(N), activate(XS)))
   U82^1(tt, N, XS) -> SPLITAT(activate(N), activate(XS))
   U82^1(tt, N, XS) -> ACTIVATE(N)
   U82^1(tt, N, XS) -> ACTIVATE(XS)
   AFTERNTH(N, XS) -> U11^1(tt, N, XS)
   FST(pair(X, Y)) -> U21^1(tt, X)
   HEAD(cons(N, XS)) -> U31^1(tt, N)
   SEL(N, XS) -> U41^1(tt, N, XS)
   SND(pair(X, Y)) -> U51^1(tt, Y)
   SPLITAT(s(N), cons(X, XS)) -> U61^1(tt, N, X, activate(XS))
   SPLITAT(s(N), cons(X, XS)) -> ACTIVATE(XS)
   TAIL(cons(N, XS)) -> U71^1(tt, activate(XS))
   TAIL(cons(N, XS)) -> ACTIVATE(XS)
   TAKE(N, XS) -> U81^1(tt, N, XS)
   ACTIVATE(n__natsFrom(X)) -> NATSFROM(X)

The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 54 less nodes.
----------------------------------------

(4)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U62^1(tt, N, X, XS) -> U63^1(tt, activate(N), activate(X), activate(XS))
   U63^1(tt, N, X, XS) -> SPLITAT(activate(N), activate(XS))
   SPLITAT(s(N), cons(X, XS)) -> U61^1(tt, N, X, activate(XS))
   U61^1(tt, N, X, XS) -> U62^1(tt, activate(N), activate(X), activate(XS))

The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(5) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U63^1(tt, N, X, XS) -> SPLITAT(activate(N), activate(XS)) at position [0] we obtained the following new rules [LPAR04]:

   (U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(natsFrom(x0), activate(y2)),U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(natsFrom(x0), activate(y2)))
   (U63^1(tt, x0, y1, y2) -> SPLITAT(x0, activate(y2)),U63^1(tt, x0, y1, y2) -> SPLITAT(x0, activate(y2)))


----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U62^1(tt, N, X, XS) -> U63^1(tt, activate(N), activate(X), activate(XS))
   SPLITAT(s(N), cons(X, XS)) -> U61^1(tt, N, X, activate(XS))
   U61^1(tt, N, X, XS) -> U62^1(tt, activate(N), activate(X), activate(XS))
   U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(natsFrom(x0), activate(y2))
   U63^1(tt, x0, y1, y2) -> SPLITAT(x0, activate(y2))

The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(natsFrom(x0), activate(y2)) at position [0] we obtained the following new rules [LPAR04]:

   (U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(cons(x0, n__natsFrom(s(x0))), activate(y2)),U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(cons(x0, n__natsFrom(s(x0))), activate(y2)))
   (U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(n__natsFrom(x0), activate(y2)),U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(n__natsFrom(x0), activate(y2)))


----------------------------------------

(8)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U62^1(tt, N, X, XS) -> U63^1(tt, activate(N), activate(X), activate(XS))
   SPLITAT(s(N), cons(X, XS)) -> U61^1(tt, N, X, activate(XS))
   U61^1(tt, N, X, XS) -> U62^1(tt, activate(N), activate(X), activate(XS))
   U63^1(tt, x0, y1, y2) -> SPLITAT(x0, activate(y2))
   U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(cons(x0, n__natsFrom(s(x0))), activate(y2))
   U63^1(tt, n__natsFrom(x0), y1, y2) -> SPLITAT(n__natsFrom(x0), activate(y2))

The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(9) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U63^1(tt, x0, y1, y2) -> SPLITAT(x0, activate(y2))
   SPLITAT(s(N), cons(X, XS)) -> U61^1(tt, N, X, activate(XS))
   U61^1(tt, N, X, XS) -> U62^1(tt, activate(N), activate(X), activate(XS))
   U62^1(tt, N, X, XS) -> U63^1(tt, activate(N), activate(X), activate(XS))

The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U63^1(tt, x0, y1, y2) -> SPLITAT(x0, activate(y2)) at position [1] we obtained the following new rules [LPAR04]:

   (U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, natsFrom(x0)),U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, natsFrom(x0)))
   (U63^1(tt, y0, y1, x0) -> SPLITAT(y0, x0),U63^1(tt, y0, y1, x0) -> SPLITAT(y0, x0))


----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SPLITAT(s(N), cons(X, XS)) -> U61^1(tt, N, X, activate(XS))
   U61^1(tt, N, X, XS) -> U62^1(tt, activate(N), activate(X), activate(XS))
   U62^1(tt, N, X, XS) -> U63^1(tt, activate(N), activate(X), activate(XS))
   U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, natsFrom(x0))
   U63^1(tt, y0, y1, x0) -> SPLITAT(y0, x0)

The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) TransformationProof (EQUIVALENT)
By narrowing [LPAR04] the rule U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, natsFrom(x0)) at position [1] we obtained the following new rules [LPAR04]:

   (U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, cons(x0, n__natsFrom(s(x0)))),U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, cons(x0, n__natsFrom(s(x0)))))
   (U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, n__natsFrom(x0)),U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, n__natsFrom(x0)))


----------------------------------------

(14)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SPLITAT(s(N), cons(X, XS)) -> U61^1(tt, N, X, activate(XS))
   U61^1(tt, N, X, XS) -> U62^1(tt, activate(N), activate(X), activate(XS))
   U62^1(tt, N, X, XS) -> U63^1(tt, activate(N), activate(X), activate(XS))
   U63^1(tt, y0, y1, x0) -> SPLITAT(y0, x0)
   U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, cons(x0, n__natsFrom(s(x0))))
   U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, n__natsFrom(x0))

The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(15) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
----------------------------------------

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U61^1(tt, N, X, XS) -> U62^1(tt, activate(N), activate(X), activate(XS))
   U62^1(tt, N, X, XS) -> U63^1(tt, activate(N), activate(X), activate(XS))
   U63^1(tt, y0, y1, x0) -> SPLITAT(y0, x0)
   SPLITAT(s(N), cons(X, XS)) -> U61^1(tt, N, X, activate(XS))
   U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, cons(x0, n__natsFrom(s(x0))))

The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(17) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   U62^1(tt, N, X, XS) -> U63^1(tt, activate(N), activate(X), activate(XS))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( U61^1_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2
POL( U62^1_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2
POL( U63^1_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 1
POL( activate_1(x_1) ) = x_1
POL( n__natsFrom_1(x_1) ) = max{0, -2}
POL( natsFrom_1(x_1) ) = 0
POL( cons_2(x_1, x_2) ) = max{0, -2}
POL( s_1(x_1) ) = x_1 + 1
POL( tt ) = 0
POL( SPLITAT_2(x_1, x_2) ) = x_1 + 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   natsFrom(X) -> n__natsFrom(X)


----------------------------------------

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   U61^1(tt, N, X, XS) -> U62^1(tt, activate(N), activate(X), activate(XS))
   U63^1(tt, y0, y1, x0) -> SPLITAT(y0, x0)
   SPLITAT(s(N), cons(X, XS)) -> U61^1(tt, N, X, activate(XS))
   U63^1(tt, y0, y1, n__natsFrom(x0)) -> SPLITAT(y0, cons(x0, n__natsFrom(s(x0))))

The TRS R consists of the following rules:

   U11(tt, N, XS) -> U12(tt, activate(N), activate(XS))
   U12(tt, N, XS) -> snd(splitAt(activate(N), activate(XS)))
   U21(tt, X) -> U22(tt, activate(X))
   U22(tt, X) -> activate(X)
   U31(tt, N) -> U32(tt, activate(N))
   U32(tt, N) -> activate(N)
   U41(tt, N, XS) -> U42(tt, activate(N), activate(XS))
   U42(tt, N, XS) -> head(afterNth(activate(N), activate(XS)))
   U51(tt, Y) -> U52(tt, activate(Y))
   U52(tt, Y) -> activate(Y)
   U61(tt, N, X, XS) -> U62(tt, activate(N), activate(X), activate(XS))
   U62(tt, N, X, XS) -> U63(tt, activate(N), activate(X), activate(XS))
   U63(tt, N, X, XS) -> U64(splitAt(activate(N), activate(XS)), activate(X))
   U64(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
   U71(tt, XS) -> U72(tt, activate(XS))
   U72(tt, XS) -> activate(XS)
   U81(tt, N, XS) -> U82(tt, activate(N), activate(XS))
   U82(tt, N, XS) -> fst(splitAt(activate(N), activate(XS)))
   afterNth(N, XS) -> U11(tt, N, XS)
   fst(pair(X, Y)) -> U21(tt, X)
   head(cons(N, XS)) -> U31(tt, N)
   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
   sel(N, XS) -> U41(tt, N, XS)
   snd(pair(X, Y)) -> U51(tt, Y)
   splitAt(0, XS) -> pair(nil, XS)
   splitAt(s(N), cons(X, XS)) -> U61(tt, N, X, activate(XS))
   tail(cons(N, XS)) -> U71(tt, activate(XS))
   take(N, XS) -> U81(tt, N, XS)
   natsFrom(X) -> n__natsFrom(X)
   activate(n__natsFrom(X)) -> natsFrom(X)
   activate(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(19) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
----------------------------------------

(20)
TRUE