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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 396 ms]
    (6) BOUNDS(1, n^1)
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 455 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 150 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 136 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 91 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 36 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 85 ms]
            (28) typed CpxTrs
            (29) RewriteLemmaProof [LOWER BOUND(ID), 93 ms]
            (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
            (32) typed CpxTrs
            (33) RewriteLemmaProof [LOWER BOUND(ID), 83 ms]
            (34) typed CpxTrs
            (35) RewriteLemmaProof [LOWER BOUND(ID), 181 ms]
            (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 127 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 150 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 115 ms]
            (42) typed CpxTrs
            (43) RewriteLemmaProof [LOWER BOUND(ID), 110 ms]
            (44) typed CpxTrs
            (45) RewriteLemmaProof [LOWER BOUND(ID), 132 ms]
            (46) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
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))
proper(U11(X1, X2, X3, X4)) -> U11(proper(X1), proper(X2), proper(X3), proper(X4))
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(tail(X)) -> tail(proper(X))
proper(take(X1, X2)) -> take(proper(X1), proper(X2))

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

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   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(tt) -> ok(tt)
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   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
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   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(tt) -> ok(tt)
   proper(0) -> ok(0)
   proper(nil) -> ok(nil)
   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) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]
transitions: 
mark0(0) -> 0
tt0() -> 0
ok0(0) -> 0
00() -> 0
nil0() -> 0
active0(0) -> 0
U110(0, 0, 0, 0) -> 1
U120(0, 0) -> 2
splitAt0(0, 0) -> 3
pair0(0, 0) -> 4
cons0(0, 0) -> 5
afterNth0(0, 0) -> 6
snd0(0) -> 7
and0(0, 0) -> 8
fst0(0) -> 9
head0(0) -> 10
natsFrom0(0) -> 11
s0(0) -> 12
sel0(0, 0) -> 13
tail0(0) -> 14
take0(0, 0) -> 15
proper0(0) -> 16
top0(0) -> 17
U111(0, 0, 0, 0) -> 18
mark1(18) -> 1
U121(0, 0) -> 19
mark1(19) -> 2
splitAt1(0, 0) -> 20
mark1(20) -> 3
pair1(0, 0) -> 21
mark1(21) -> 4
cons1(0, 0) -> 22
mark1(22) -> 5
afterNth1(0, 0) -> 23
mark1(23) -> 6
snd1(0) -> 24
mark1(24) -> 7
and1(0, 0) -> 25
mark1(25) -> 8
fst1(0) -> 26
mark1(26) -> 9
head1(0) -> 27
mark1(27) -> 10
natsFrom1(0) -> 28
mark1(28) -> 11
s1(0) -> 29
mark1(29) -> 12
sel1(0, 0) -> 30
mark1(30) -> 13
tail1(0) -> 31
mark1(31) -> 14
take1(0, 0) -> 32
mark1(32) -> 15
tt1() -> 33
ok1(33) -> 16
01() -> 34
ok1(34) -> 16
nil1() -> 35
ok1(35) -> 16
U111(0, 0, 0, 0) -> 36
ok1(36) -> 1
U121(0, 0) -> 37
ok1(37) -> 2
splitAt1(0, 0) -> 38
ok1(38) -> 3
pair1(0, 0) -> 39
ok1(39) -> 4
cons1(0, 0) -> 40
ok1(40) -> 5
afterNth1(0, 0) -> 41
ok1(41) -> 6
snd1(0) -> 42
ok1(42) -> 7
and1(0, 0) -> 43
ok1(43) -> 8
fst1(0) -> 44
ok1(44) -> 9
head1(0) -> 45
ok1(45) -> 10
natsFrom1(0) -> 46
ok1(46) -> 11
s1(0) -> 47
ok1(47) -> 12
sel1(0, 0) -> 48
ok1(48) -> 13
tail1(0) -> 49
ok1(49) -> 14
take1(0, 0) -> 50
ok1(50) -> 15
proper1(0) -> 51
top1(51) -> 17
active1(0) -> 52
top1(52) -> 17
mark1(18) -> 18
mark1(18) -> 36
mark1(19) -> 19
mark1(19) -> 37
mark1(20) -> 20
mark1(20) -> 38
mark1(21) -> 21
mark1(21) -> 39
mark1(22) -> 22
mark1(22) -> 40
mark1(23) -> 23
mark1(23) -> 41
mark1(24) -> 24
mark1(24) -> 42
mark1(25) -> 25
mark1(25) -> 43
mark1(26) -> 26
mark1(26) -> 44
mark1(27) -> 27
mark1(27) -> 45
mark1(28) -> 28
mark1(28) -> 46
mark1(29) -> 29
mark1(29) -> 47
mark1(30) -> 30
mark1(30) -> 48
mark1(31) -> 31
mark1(31) -> 49
mark1(32) -> 32
mark1(32) -> 50
ok1(33) -> 51
ok1(34) -> 51
ok1(35) -> 51
ok1(36) -> 18
ok1(36) -> 36
ok1(37) -> 19
ok1(37) -> 37
ok1(38) -> 20
ok1(38) -> 38
ok1(39) -> 21
ok1(39) -> 39
ok1(40) -> 22
ok1(40) -> 40
ok1(41) -> 23
ok1(41) -> 41
ok1(42) -> 24
ok1(42) -> 42
ok1(43) -> 25
ok1(43) -> 43
ok1(44) -> 26
ok1(44) -> 44
ok1(45) -> 27
ok1(45) -> 45
ok1(46) -> 28
ok1(46) -> 46
ok1(47) -> 29
ok1(47) -> 47
ok1(48) -> 30
ok1(48) -> 48
ok1(49) -> 31
ok1(49) -> 49
ok1(50) -> 32
ok1(50) -> 50
active2(33) -> 53
top2(53) -> 17
active2(34) -> 53
active2(35) -> 53

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

(6)
BOUNDS(1, n^1)

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

(7) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


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

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
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

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

(11) OrderProof (LOWER BOUND(ID))
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

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

(12)
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

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
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 in Omega(n5_0)

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)) ->_R^Omega(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 in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following 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

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

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^1, INF)

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

(18)
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 in Omega(n5_0)


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

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)

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, +(n1672_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)


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

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)

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, +(n3844_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)


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

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)

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, +(n6322_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(24)
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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)


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

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)

Induction Base:
snd(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
snd(gen_tt:mark:0':nil:ok3_0(+(1, +(n8907_0, 1)))) ->_R^Omega(1)
mark(snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)


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

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)

Induction Base:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, +(n10038_0, 1)))) ->_R^Omega(1)
mark(natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(28)
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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)


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

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

(29) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)

Induction Base:
s(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
s(gen_tt:mark:0':nil:ok3_0(+(1, +(n11270_0, 1)))) ->_R^Omega(1)
mark(s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(30)
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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)


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

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0))) -> *4_0, rt in Omega(n12603_0)

Induction Base:
head(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
head(gen_tt:mark:0':nil:ok3_0(+(1, +(n12603_0, 1)))) ->_R^Omega(1)
mark(head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0))) -> *4_0, rt in Omega(n12603_0)


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))
Proved the following rewrite lemma:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n14037_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n14037_0)

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, +(n14037_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n14037_0)), gen_tt:mark:0':nil:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(34)
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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0))) -> *4_0, rt in Omega(n12603_0)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n14037_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n14037_0)


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

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

(35) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17959_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 in Omega(n17959_0)

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, +(n17959_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)) ->_R^Omega(1)
mark(U11(gen_tt:mark:0':nil:ok3_0(+(1, n17959_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 in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(36)
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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0))) -> *4_0, rt in Omega(n12603_0)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n14037_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n14037_0)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17959_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 in Omega(n17959_0)


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

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26676_0))) -> *4_0, rt in Omega(n26676_0)

Induction Base:
fst(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
fst(gen_tt:mark:0':nil:ok3_0(+(1, +(n26676_0, 1)))) ->_R^Omega(1)
mark(fst(gen_tt:mark:0':nil:ok3_0(+(1, n26676_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0))) -> *4_0, rt in Omega(n12603_0)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n14037_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n14037_0)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17959_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 in Omega(n17959_0)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26676_0))) -> *4_0, rt in Omega(n26676_0)


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

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
and(gen_tt:mark:0':nil:ok3_0(+(1, n28609_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n28609_0)

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, +(n28609_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(and(gen_tt:mark:0':nil:ok3_0(+(1, n28609_0)), gen_tt:mark:0':nil:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(40)
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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0))) -> *4_0, rt in Omega(n12603_0)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n14037_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n14037_0)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17959_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 in Omega(n17959_0)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26676_0))) -> *4_0, rt in Omega(n26676_0)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28609_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n28609_0)


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

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sel(gen_tt:mark:0':nil:ok3_0(+(1, n33348_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n33348_0)

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, +(n33348_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(sel(gen_tt:mark:0':nil:ok3_0(+(1, n33348_0)), gen_tt:mark:0':nil:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(42)
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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0))) -> *4_0, rt in Omega(n12603_0)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n14037_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n14037_0)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17959_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 in Omega(n17959_0)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26676_0))) -> *4_0, rt in Omega(n26676_0)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28609_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n28609_0)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n33348_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n33348_0)


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

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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
tail(gen_tt:mark:0':nil:ok3_0(+(1, n38592_0))) -> *4_0, rt in Omega(n38592_0)

Induction Base:
tail(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
tail(gen_tt:mark:0':nil:ok3_0(+(1, +(n38592_0, 1)))) ->_R^Omega(1)
mark(tail(gen_tt:mark:0':nil:ok3_0(+(1, n38592_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0))) -> *4_0, rt in Omega(n12603_0)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n14037_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n14037_0)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17959_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 in Omega(n17959_0)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26676_0))) -> *4_0, rt in Omega(n26676_0)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28609_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n28609_0)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n33348_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n33348_0)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n38592_0))) -> *4_0, rt in Omega(n38592_0)


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

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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
take(gen_tt:mark:0':nil:ok3_0(+(1, n40926_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n40926_0)

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, +(n40926_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) ->_R^Omega(1)
mark(take(gen_tt:mark:0':nil:ok3_0(+(1, n40926_0)), gen_tt:mark:0':nil:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(46)
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 in Omega(n5_0)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n1672_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n1672_0)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n3844_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n3844_0)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n6322_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n6322_0)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n8907_0))) -> *4_0, rt in Omega(n8907_0)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n10038_0))) -> *4_0, rt in Omega(n10038_0)
s(gen_tt:mark:0':nil:ok3_0(+(1, n11270_0))) -> *4_0, rt in Omega(n11270_0)
head(gen_tt:mark:0':nil:ok3_0(+(1, n12603_0))) -> *4_0, rt in Omega(n12603_0)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n14037_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n14037_0)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n17959_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 in Omega(n17959_0)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n26676_0))) -> *4_0, rt in Omega(n26676_0)
and(gen_tt:mark:0':nil:ok3_0(+(1, n28609_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n28609_0)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n33348_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n33348_0)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n38592_0))) -> *4_0, rt in Omega(n38592_0)
take(gen_tt:mark:0':nil:ok3_0(+(1, n40926_0)), gen_tt:mark:0':nil:ok3_0(b)) -> *4_0, rt in Omega(n40926_0)


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