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

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

U11(tt, N, X, XS) → U12(splitAt(activate(N), activate(XS)), activate(X))
U12(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
afterNth(N, XS) → snd(splitAt(N, XS))
and(tt, X) → activate(X)
fst(pair(X, Y)) → X
head(cons(N, XS)) → N
natsFrom(N) → cons(N, n__natsFrom(s(N)))
sel(N, XS) → head(afterNth(N, XS))
snd(pair(X, Y)) → Y
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U11(tt, N, X, activate(XS))
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(splitAt(N, XS))
natsFrom(X) → n__natsFrom(X)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
AND(tt, z0) → c3(ACTIVATE(z0))
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
ACTIVATE(n__natsFrom(z0)) → c14(NATSFROM(z0))

S tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
AND(tt, z0) → c3(ACTIVATE(z0))
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
ACTIVATE(n__natsFrom(z0)) → c14(NATSFROM(z0))

K tuples:none
Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', U12', AFTERNTH, AND, SEL, SPLITAT, TAIL, TAKE, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c8, c11, c12, c13, c14

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
AND(tt, z0) → c3(ACTIVATE(z0))
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
ACTIVATE(n__natsFrom(z0)) → c14(NATSFROM(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1))

S tuples:

U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), SPLITAT(activate(z0), activate(z2)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1))

K tuples:none
Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', AFTERNTH, SEL, SPLITAT, TAKE

Compound Symbols:

c, c2, c8, c11, c13

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 8 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))
SEL(z0, z1) → c8(AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))

S tuples:

U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))
SEL(z0, z1) → c8(AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))

K tuples:none
Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', AFTERNTH, SEL, SPLITAT, TAKE

Compound Symbols:

c, c2, c8, c11, c13

(7) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

SEL(z0, z1) → c8(AFTERNTH(z0, z1))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))
SEL(z0, z1) → c8(AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))

S tuples:

U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))

K tuples:

SEL(z0, z1) → c8(AFTERNTH(z0, z1))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))

Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', AFTERNTH, SEL, SPLITAT, TAKE

Compound Symbols:

c, c2, c8, c11, c13

(9) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))

We considered the (Usable) Rules:

activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)

And the Tuples:

U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))
SEL(z0, z1) → c8(AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(AFTERNTH(x₁, x₂)) = [5]x₁ + [3]x₂   
POL(SEL(x₁, x₂)) = [4] + [5]x₁ + [3]x₂   
POL(SPLITAT(x₁, x₂)) = x₁   
POL(TAKE(x₁, x₂)) = [5] + [5]x₁ + [4]x₂   
POL(U11'(x₁, x₂, x₃, x₄)) = x₁ + x₂   
POL(activate(x₁)) = x₁   
POL(c(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁   
POL(n__natsFrom(x₁)) = [4] + x₁   
POL(natsFrom(x₁)) = [4] + x₁   
POL(s(x₁)) = [2] + x₁   
POL(tt) = [1]   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U11(tt, z0, z1, z2) → U12(splitAt(activate(z0), activate(z2)), activate(z1))
U12(pair(z0, z1), z2) → pair(cons(activate(z2), z0), z1)
afterNth(z0, z1) → snd(splitAt(z0, z1))
and(tt, z0) → activate(z0)
fst(pair(z0, z1)) → z0
head(cons(z0, z1)) → z0
natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
sel(z0, z1) → head(afterNth(z0, z1))
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → U11(tt, z0, z1, activate(z2))
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0

Tuples:

U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))
SEL(z0, z1) → c8(AFTERNTH(z0, z1))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))

S tuples:none
K tuples:

SEL(z0, z1) → c8(AFTERNTH(z0, z1))
TAKE(z0, z1) → c13(SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c2(SPLITAT(z0, z1))
U11'(tt, z0, z1, z2) → c(SPLITAT(activate(z0), activate(z2)))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)))

Defined Rule Symbols:

U11, U12, afterNth, and, fst, head, natsFrom, sel, snd, splitAt, tail, take, activate

Defined Pair Symbols:

U11', AFTERNTH, SEL, SPLITAT, TAKE

Compound Symbols:

c, c2, c8, c11, c13

(11) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(12) BOUNDS(O(1), O(1))