KILLED



    


Runtime Complexity (full) proof of /tmp/tmpfJwLrA/LISTUTILITIES_nosorts_C.xml









The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

0 CpxTRS


↳1 DecreasingLoopProof (⇔, 10.2 s)


↳2 BOUNDS(n^1, INF)


↳3 RenamingProof (⇔, 0 ms)


↳4 CpxRelTRS


    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)


    ↳6 typed CpxTrs


    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)


    ↳8 typed CpxTrs


    ↳9 RewriteLemmaProof (LOWER BOUND(ID), 413 ms)


    ↳10 BEST


        ↳11 typed CpxTrs


            ↳12 RewriteLemmaProof (LOWER BOUND(ID), 116 ms)


            ↳13 BEST


                ↳14 typed CpxTrs


                    ↳15 RewriteLemmaProof (LOWER BOUND(ID), 115 ms)


                    ↳16 BEST


                        ↳17 typed CpxTrs


                            ↳18 RewriteLemmaProof (LOWER BOUND(ID), 116 ms)


                            ↳19 BEST


                                ↳20 typed CpxTrs


                                    ↳21 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)


                                    ↳22 BEST


                                        ↳23 typed CpxTrs


                                            ↳24 RewriteLemmaProof (LOWER BOUND(ID), 18 ms)


                                            ↳25 BEST


                                                ↳26 typed CpxTrs


                                                    ↳27 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)


                                                    ↳28 BEST


                                                        ↳29 typed CpxTrs


                                                            ↳30 RewriteLemmaProof (LOWER BOUND(ID), 21 ms)


                                                            ↳31 BEST


                                                                ↳32 typed CpxTrs


                                                                    ↳33 RewriteLemmaProof (LOWER BOUND(ID), 144 ms)


                                                                    ↳34 BEST


                                                                        ↳35 typed CpxTrs


                                                                            ↳36 RewriteLemmaProof (LOWER BOUND(ID), 172 ms)


                                                                            ↳37 BEST


                                                                                ↳38 typed CpxTrs


                                                                                    ↳39 RewriteLemmaProof (LOWER BOUND(ID), 68 ms)


                                                                                    ↳40 BEST


                                                                                        ↳41 typed CpxTrs


                                                                                            ↳42 RewriteLemmaProof (LOWER BOUND(ID), 92 ms)


                                                                                            ↳43 BEST


                                                                                                ↳44 typed CpxTrs


                                                                                                    ↳45 RewriteLemmaProof (LOWER BOUND(ID), 56 ms)


                                                                                                    ↳46 BEST


                                                                                                        ↳47 typed CpxTrs


                                                                                                            ↳48 RewriteLemmaProof (LOWER BOUND(ID), 106 ms)


                                                                                                            ↳49 BEST


                                                                                                                ↳50 typed CpxTrs


                                                                                                                    ↳51 RewriteLemmaProof (LOWER BOUND(ID), 93 ms)


                                                                                                                    ↳52 BEST


                                                                                                                        ↳53 typed CpxTrs


                                                                                                                        ↳54 typed CpxTrs


                                                                                                                ↳55 typed CpxTrs


                                                                                                        ↳56 typed CpxTrs


                                                                                                ↳57 typed CpxTrs


                                                                                        ↳58 typed CpxTrs


                                                                                ↳59 typed CpxTrs


                                                                        ↳60 typed CpxTrs


                                                                ↳61 typed CpxTrs


                                                        ↳62 typed CpxTrs


                                                ↳63 typed CpxTrs


                                        ↳64 typed CpxTrs


                                ↳65 typed CpxTrs


                        ↳66 typed CpxTrs


                ↳67 typed CpxTrs


        ↳68 typed CpxTrs









(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))

active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))

active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))

active(and(tt, X)) → mark(X)

active(fst(pair(X, Y))) → mark(X)

active(head(cons(N, XS))) → mark(N)

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))

active(sel(N, XS)) → mark(head(afterNth(N, XS)))

active(snd(pair(X, Y))) → mark(Y)

active(splitAt(0, XS)) → mark(pair(nil, XS))

active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))

active(tail(cons(N, XS))) → mark(XS)

active(take(N, XS)) → mark(fst(splitAt(N, XS)))

active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)

active(U12(X1, X2)) → U12(active(X1), X2)

active(splitAt(X1, X2)) → splitAt(active(X1), X2)

active(splitAt(X1, X2)) → splitAt(X1, active(X2))

active(pair(X1, X2)) → pair(active(X1), X2)

active(pair(X1, X2)) → pair(X1, active(X2))

active(cons(X1, X2)) → cons(active(X1), X2)

active(afterNth(X1, X2)) → afterNth(active(X1), X2)

active(afterNth(X1, X2)) → afterNth(X1, active(X2))

active(snd(X)) → snd(active(X))

active(and(X1, X2)) → and(active(X1), X2)

active(fst(X)) → fst(active(X))

active(head(X)) → head(active(X))

active(natsFrom(X)) → natsFrom(active(X))

active(s(X)) → s(active(X))

active(sel(X1, X2)) → sel(active(X1), X2)

active(sel(X1, X2)) → sel(X1, active(X2))

active(tail(X)) → tail(active(X))

active(take(X1, X2)) → take(active(X1), X2)

active(take(X1, X2)) → take(X1, active(X2))

U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))

U12(mark(X1), X2) → mark(U12(X1, X2))

splitAt(mark(X1), X2) → mark(splitAt(X1, X2))

splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))

pair(mark(X1), X2) → mark(pair(X1, X2))

pair(X1, mark(X2)) → mark(pair(X1, X2))

cons(mark(X1), X2) → mark(cons(X1, X2))

afterNth(mark(X1), X2) → mark(afterNth(X1, X2))

afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))

snd(mark(X)) → mark(snd(X))

and(mark(X1), X2) → mark(and(X1, X2))

fst(mark(X)) → mark(fst(X))

head(mark(X)) → mark(head(X))

natsFrom(mark(X)) → mark(natsFrom(X))

s(mark(X)) → mark(s(X))

sel(mark(X1), X2) → mark(sel(X1, X2))

sel(X1, mark(X2)) → mark(sel(X1, X2))

tail(mark(X)) → mark(tail(X))

take(mark(X1), X2) → mark(take(X1, X2))

take(X1, mark(X2)) → mark(take(X1, X2))

proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))

proper(tt) → ok(tt)

proper(U12(X1, X2)) → U12(proper(X1), proper(X2))

proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))

proper(pair(X1, X2)) → pair(proper(X1), proper(X2))

proper(cons(X1, X2)) → cons(proper(X1), proper(X2))

proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))

proper(snd(X)) → snd(proper(X))

proper(and(X1, X2)) → and(proper(X1), proper(X2))

proper(fst(X)) → fst(proper(X))

proper(head(X)) → head(proper(X))

proper(natsFrom(X)) → natsFrom(proper(X))

proper(s(X)) → s(proper(X))

proper(sel(X1, X2)) → sel(proper(X1), proper(X2))

proper(0) → ok(0)

proper(nil) → ok(nil)

proper(tail(X)) → tail(proper(X))

proper(take(X1, X2)) → take(proper(X1), proper(X2))

U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))

U12(ok(X1), ok(X2)) → ok(U12(X1, X2))

splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))

pair(ok(X1), ok(X2)) → ok(pair(X1, X2))

cons(ok(X1), ok(X2)) → ok(cons(X1, X2))

afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))

snd(ok(X)) → ok(snd(X))

and(ok(X1), ok(X2)) → ok(and(X1, X2))

fst(ok(X)) → ok(fst(X))

head(ok(X)) → ok(head(X))

natsFrom(ok(X)) → ok(natsFrom(X))

s(ok(X)) → ok(s(X))

sel(ok(X1), ok(X2)) → ok(sel(X1, X2))

tail(ok(X)) → ok(tail(X))

take(ok(X1), ok(X2)) → ok(take(X1, X2))

top(mark(X)) → top(proper(X))

top(ok(X)) → top(active(X))

Rewrite Strategy: FULL




(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
U11(mark(X1), X2, X3, X4) →+ mark(U11(X1, X2, X3, X4))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].





(2) BOUNDS(n^1, INF)




(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.




(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))

active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))

active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))

active(and(tt, X)) → mark(X)

active(fst(pair(X, Y))) → mark(X)

active(head(cons(N, XS))) → mark(N)

active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))

active(sel(N, XS)) → mark(head(afterNth(N, XS)))

active(snd(pair(X, Y))) → mark(Y)

active(splitAt(0', XS)) → mark(pair(nil, XS))

active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))

active(tail(cons(N, XS))) → mark(XS)

active(take(N, XS)) → mark(fst(splitAt(N, XS)))

active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)

active(U12(X1, X2)) → U12(active(X1), X2)

active(splitAt(X1, X2)) → splitAt(active(X1), X2)

active(splitAt(X1, X2)) → splitAt(X1, active(X2))

active(pair(X1, X2)) → pair(active(X1), X2)

active(pair(X1, X2)) → pair(X1, active(X2))

active(cons(X1, X2)) → cons(active(X1), X2)

active(afterNth(X1, X2)) → afterNth(active(X1), X2)

active(afterNth(X1, X2)) → afterNth(X1, active(X2))

active(snd(X)) → snd(active(X))

active(and(X1, X2)) → and(active(X1), X2)

active(fst(X)) → fst(active(X))

active(head(X)) → head(active(X))

active(natsFrom(X)) → natsFrom(active(X))

active(s(X)) → s(active(X))

active(sel(X1, X2)) → sel(active(X1), X2)

active(sel(X1, X2)) → sel(X1, active(X2))

active(tail(X)) → tail(active(X))

active(take(X1, X2)) → take(active(X1), X2)

active(take(X1, X2)) → take(X1, active(X2))

U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))

U12(mark(X1), X2) → mark(U12(X1, X2))

splitAt(mark(X1), X2) → mark(splitAt(X1, X2))

splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))

pair(mark(X1), X2) → mark(pair(X1, X2))

pair(X1, mark(X2)) → mark(pair(X1, X2))

cons(mark(X1), X2) → mark(cons(X1, X2))

afterNth(mark(X1), X2) → mark(afterNth(X1, X2))

afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))

snd(mark(X)) → mark(snd(X))

and(mark(X1), X2) → mark(and(X1, X2))

fst(mark(X)) → mark(fst(X))

head(mark(X)) → mark(head(X))

natsFrom(mark(X)) → mark(natsFrom(X))

s(mark(X)) → mark(s(X))

sel(mark(X1), X2) → mark(sel(X1, X2))

sel(X1, mark(X2)) → mark(sel(X1, X2))

tail(mark(X)) → mark(tail(X))

take(mark(X1), X2) → mark(take(X1, X2))

take(X1, mark(X2)) → mark(take(X1, X2))

proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))

proper(tt) → ok(tt)

proper(U12(X1, X2)) → U12(proper(X1), proper(X2))

proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))

proper(pair(X1, X2)) → pair(proper(X1), proper(X2))

proper(cons(X1, X2)) → cons(proper(X1), proper(X2))

proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))

proper(snd(X)) → snd(proper(X))

proper(and(X1, X2)) → and(proper(X1), proper(X2))

proper(fst(X)) → fst(proper(X))

proper(head(X)) → head(proper(X))

proper(natsFrom(X)) → natsFrom(proper(X))

proper(s(X)) → s(proper(X))

proper(sel(X1, X2)) → sel(proper(X1), proper(X2))

proper(0') → ok(0')

proper(nil) → ok(nil)

proper(tail(X)) → tail(proper(X))

proper(take(X1, X2)) → take(proper(X1), proper(X2))

U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))

U12(ok(X1), ok(X2)) → ok(U12(X1, X2))

splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))

pair(ok(X1), ok(X2)) → ok(pair(X1, X2))

cons(ok(X1), ok(X2)) → ok(cons(X1, X2))

afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))

snd(ok(X)) → ok(snd(X))

and(ok(X1), ok(X2)) → ok(and(X1, X2))

fst(ok(X)) → ok(fst(X))

head(ok(X)) → ok(head(X))

natsFrom(ok(X)) → ok(natsFrom(X))

s(ok(X)) → ok(s(X))

sel(ok(X1), ok(X2)) → ok(sel(X1, X2))

tail(ok(X)) → ok(tail(X))

take(ok(X1), ok(X2)) → ok(take(X1, X2))

top(mark(X)) → top(proper(X))

top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL




(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.




(6) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok




(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, U12, splitAt, pair, cons, snd, natsFrom, s, head, afterNth, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U12 < active
splitAt < active
pair < active
cons < active
snd < active
natsFrom < active
s < active
head < active
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
U12 < proper
splitAt < proper
pair < proper
cons < proper
snd < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(8) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U12, active, splitAt, pair, cons, snd, natsFrom, s, head, afterNth, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U12 < active
splitAt < active
pair < active
cons < active
snd < active
natsFrom < active
s < active
head < active
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
U12 < proper
splitAt < proper
pair < proper
cons < proper
snd < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Induction Base:
U12(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))
Induction Step:
U12(gen_tt:mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(10) Complex Obligation (BEST)




(11) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
splitAt, active, pair, cons, snd, natsFrom, s, head, afterNth, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
splitAt < active
pair < active
cons < active
snd < active
natsFrom < active
s < active
head < active
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
splitAt < proper
pair < proper
cons < proper
snd < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
Induction Base:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))
Induction Step:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, +(n1598_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(13) Complex Obligation (BEST)




(14) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
pair, active, cons, snd, natsFrom, s, head, afterNth, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
pair < active
cons < active
snd < active
natsFrom < active
s < active
head < active
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
pair < proper
cons < proper
snd < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
Induction Base:
pair(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))
Induction Step:
pair(gen_tt:mark:0':nil:ok3_0(+(1, +(n3694_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(16) Complex Obligation (BEST)




(17) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
cons, active, snd, natsFrom, s, head, afterNth, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
cons < active
snd < active
natsFrom < active
s < active
head < active
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
cons < proper
snd < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
Induction Base:
cons(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))
Induction Step:
cons(gen_tt:mark:0':nil:ok3_0(+(1, +(n6094_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(19) Complex Obligation (BEST)




(20) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
snd, active, natsFrom, s, head, afterNth, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
snd < active
natsFrom < active
s < active
head < active
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
snd < proper
natsFrom < proper
s < proper
head < proper
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
Induction Base:
snd(gen_tt:mark:0':nil:ok3_0(+(1, 0)))
Induction Step:
snd(gen_tt:mark:0':nil:ok3_0(+(1, +(n8599_0, 1)))) →RΩ(1)
mark(snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(22) Complex Obligation (BEST)




(23) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
natsFrom, active, s, head, afterNth, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
natsFrom < active
s < active
head < active
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
natsFrom < proper
s < proper
head < proper
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
Induction Base:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, 0)))
Induction Step:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, +(n9730_0, 1)))) →RΩ(1)
mark(natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(25) Complex Obligation (BEST)




(26) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
s, active, head, afterNth, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
s < active
head < active
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
s < proper
head < proper
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(27) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
Induction Base:
s(gen_tt:mark:0':nil:ok3_0(+(1, 0)))
Induction Step:
s(gen_tt:mark:0':nil:ok3_0(+(1, +(n10962_0, 1)))) →RΩ(1)
mark(s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(28) Complex Obligation (BEST)




(29) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
head, active, afterNth, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
head < active
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
head < proper
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(30) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
Induction Base:
head(gen_tt:mark:0':nil:ok3_0(+(1, 0)))
Induction Step:
head(gen_tt:mark:0':nil:ok3_0(+(1, +(n12295_0, 1)))) →RΩ(1)
mark(head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(31) Complex Obligation (BEST)




(32) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
afterNth, active, U11, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
afterNth < active
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
afterNth < proper
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(33) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
Induction Base:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))
Induction Step:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, +(n13729_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(34) Complex Obligation (BEST)




(35) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
U11, active, fst, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
U11 < active
fst < active
and < active
sel < active
tail < active
take < active
active < top
U11 < proper
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(36) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
Induction Base:
U11(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))
Induction Step:
U11(gen_tt:mark:0':nil:ok3_0(+(1, +(n17561_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) →RΩ(1)
mark(U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(37) Complex Obligation (BEST)




(38) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
fst, active, and, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
fst < active
and < active
sel < active
tail < active
take < active
active < top
fst < proper
and < proper
sel < proper
tail < proper
take < proper
proper < top




(39) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
Induction Base:
fst(gen_tt:mark:0':nil:ok3_0(+(1, 0)))
Induction Step:
fst(gen_tt:mark:0':nil:ok3_0(+(1, +(n26418_0, 1)))) →RΩ(1)
mark(fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(40) Complex Obligation (BEST)




(41) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
and, active, sel, tail, take, proper, top
They will be analysed ascendingly in the following order:
and < active
sel < active
tail < active
take < active
active < top
and < proper
sel < proper
tail < proper
take < proper
proper < top




(42) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n283510)
Induction Base:
and(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))
Induction Step:
and(gen_tt:mark:0':nil:ok3_0(+(1, +(n28351_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(43) Complex Obligation (BEST)




(44) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n283510)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
sel, active, tail, take, proper, top
They will be analysed ascendingly in the following order:
sel < active
tail < active
take < active
active < top
sel < proper
tail < proper
take < proper
proper < top




(45) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel(gen_tt:mark:0':nil:ok3_0(+(1, n32994_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n329940)
Induction Base:
sel(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))
Induction Step:
sel(gen_tt:mark:0':nil:ok3_0(+(1, +(n32994_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(sel(gen_tt:mark:0':nil:ok3_0(+(1, n32994_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(46) Complex Obligation (BEST)




(47) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n283510)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n32994_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n329940)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
tail, active, take, proper, top
They will be analysed ascendingly in the following order:
tail < active
take < active
active < top
tail < proper
take < proper
proper < top




(48) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
tail(gen_tt:mark:0':nil:ok3_0(+(1, n38140_0))) → *4_0, rt ∈ Ω(n381400)
Induction Base:
tail(gen_tt:mark:0':nil:ok3_0(+(1, 0)))
Induction Step:
tail(gen_tt:mark:0':nil:ok3_0(+(1, +(n38140_0, 1)))) →RΩ(1)
mark(tail(gen_tt:mark:0':nil:ok3_0(+(1, n38140_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(49) Complex Obligation (BEST)




(50) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n283510)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n32994_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n329940)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n38140_0))) → *4_0, rt ∈ Ω(n381400)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
take, active, proper, top
They will be analysed ascendingly in the following order:
take < active
active < top
take < proper
proper < top




(51) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
take(gen_tt:mark:0':nil:ok3_0(+(1, n40474_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n404740)
Induction Base:
take(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))
Induction Step:
take(gen_tt:mark:0':nil:ok3_0(+(1, +(n40474_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →RΩ(1)
mark(take(gen_tt:mark:0':nil:ok3_0(+(1, n40474_0)), gen_tt:mark:0':nil:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).




(52) Complex Obligation (BEST)




(53) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n283510)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n32994_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n329940)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n38140_0))) → *4_0, rt ∈ Ω(n381400)
take(gen_tt:mark:0':nil:ok3_0(+(1, n40474_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n404740)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
The following defined symbols remain to be analysed:
active, proper, top
They will be analysed ascendingly in the following order:
active < top
proper < top




(54) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n283510)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n32994_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n329940)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n38140_0))) → *4_0, rt ∈ Ω(n381400)
take(gen_tt:mark:0':nil:ok3_0(+(1, n40474_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n404740)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(55) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n283510)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n32994_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n329940)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n38140_0))) → *4_0, rt ∈ Ω(n381400)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(56) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n283510)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n32994_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n329940)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(57) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28351_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n283510)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(58) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26418_0))) → *4_0, rt ∈ Ω(n264180)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(59) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17561_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n175610)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(60) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n13729_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n137290)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(61) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12295_0))) → *4_0, rt ∈ Ω(n122950)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(62) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
s(gen_tt:mark:0':nil:ok3_0(+(1, n10962_0))) → *4_0, rt ∈ Ω(n109620)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(63) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n9730_0))) → *4_0, rt ∈ Ω(n97300)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(64) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8599_0))) → *4_0, rt ∈ Ω(n85990)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(65) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6094_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n60940)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(66) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3694_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n36940)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(67) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1598_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n15980)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.




(68) Obligation:
TRS:
Rules:
active(U11(tt, N, X, XS)) → mark(U12(splitAt(N, XS), X))
active(U12(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(afterNth(N, XS)) → mark(snd(splitAt(N, XS)))
active(and(tt, X)) → mark(X)
active(fst(pair(X, Y))) → mark(X)
active(head(cons(N, XS))) → mark(N)
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(head(afterNth(N, XS)))
active(snd(pair(X, Y))) → mark(Y)
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U11(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(XS)
active(take(N, XS)) → mark(fst(splitAt(N, XS)))
active(U11(X1, X2, X3, X4)) → U11(active(X1), X2, X3, X4)
active(U12(X1, X2)) → U12(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(active(X1), X2)
active(afterNth(X1, X2)) → afterNth(X1, active(X2))
active(snd(X)) → snd(active(X))
active(and(X1, X2)) → and(active(X1), X2)
active(fst(X)) → fst(active(X))
active(head(X)) → head(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3, X4) → mark(U11(X1, X2, X3, X4))
U12(mark(X1), X2) → mark(U12(X1, X2))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
afterNth(mark(X1), X2) → mark(afterNth(X1, X2))
afterNth(X1, mark(X2)) → mark(afterNth(X1, X2))
snd(mark(X)) → mark(snd(X))
and(mark(X1), X2) → mark(and(X1, X2))
fst(mark(X)) → mark(fst(X))
head(mark(X)) → mark(head(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3, X4)) → U11(proper(X1), proper(X2), proper(X3), proper(X4))
proper(tt) → ok(tt)
proper(U12(X1, X2)) → U12(proper(X1), proper(X2))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(snd(X)) → snd(proper(X))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(fst(X)) → fst(proper(X))
proper(head(X)) → head(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U11(X1, X2, X3, X4))
U12(ok(X1), ok(X2)) → ok(U12(X1, X2))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
snd(ok(X)) → ok(snd(X))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
fst(ok(X)) → ok(fst(X))
head(ok(X)) → ok(head(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
and :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok
Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))
No more defined symbols left to analyse.