### (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
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
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)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Rewrite Strategy: INNERMOST

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

Converted Cpx (relative) TRS 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
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
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))
s(z0) → n__s(z0)
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
activate(n__s(z0)) → s(activate(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))
FST(pair(z0, z1)) → c4
NATSFROM(z0) → c6
NATSFROM(z0) → c7
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SND(pair(z0, z1)) → c9
SPLITAT(0, z0) → c10
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))
S(z0) → c14
ACTIVATE(n__natsFrom(z0)) → c15(NATSFROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c17
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))
FST(pair(z0, z1)) → c4
NATSFROM(z0) → c6
NATSFROM(z0) → c7
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
SND(pair(z0, z1)) → c9
SPLITAT(0, z0) → c10
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))
S(z0) → c14
ACTIVATE(n__natsFrom(z0)) → c15(NATSFROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c17
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

U11', U12', AFTERNTH, AND, FST, HEAD, NATSFROM, SEL, SND, SPLITAT, TAIL, TAKE, S, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17

### (3) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

AND(tt, z0) → c3(ACTIVATE(z0))
SPLITAT(s(z0), cons(z1, z2)) → c11(U11'(tt, z0, z1, activate(z2)), ACTIVATE(z2))
TAIL(cons(z0, z1)) → c12(ACTIVATE(z1))
Removed 11 trailing nodes:

SPLITAT(0, z0) → c10
ACTIVATE(z0) → c17
S(z0) → c14
AFTERNTH(z0, z1) → c2(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
FST(pair(z0, z1)) → c4
SND(pair(z0, z1)) → c9
TAKE(z0, z1) → c13(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
NATSFROM(z0) → c6
SEL(z0, z1) → c8(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
NATSFROM(z0) → c7

### (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
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
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))
s(z0) → n__s(z0)
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
activate(n__s(z0)) → s(activate(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))
ACTIVATE(n__natsFrom(z0)) → c15(NATSFROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(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))
ACTIVATE(n__natsFrom(z0)) → c15(NATSFROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

U11', U12', ACTIVATE

Compound Symbols:

c, c1, c15, c16

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

Removed 3 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
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
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))
s(z0) → n__s(z0)
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
S tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
U11'(tt, z0, z1, z2) → c(U12'(splitAt(activate(z0), activate(z2)), activate(z1)), ACTIVATE(z0), ACTIVATE(z2), ACTIVATE(z1))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

U12', U11', ACTIVATE

Compound Symbols:

c1, c, c15, c16

### (7) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

### (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
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
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))
s(z0) → n__s(z0)
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
U11'(tt, z0, z1, z2) → c2(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(ACTIVATE(z2))
U11'(tt, z0, z1, z2) → c2(ACTIVATE(z1))
S tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
U11'(tt, z0, z1, z2) → c2(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(ACTIVATE(z2))
U11'(tt, z0, z1, z2) → c2(ACTIVATE(z1))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

U12', ACTIVATE, U11'

Compound Symbols:

c1, c15, c16, c2

### (9) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

U11'(tt, z0, z1, z2) → c2(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(ACTIVATE(z2))
U11'(tt, z0, z1, z2) → c2(ACTIVATE(z1))

### (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
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
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))
s(z0) → n__s(z0)
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
S tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
K tuples:none
Defined Rule Symbols:

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

Defined Pair Symbols:

U12', ACTIVATE, U11'

Compound Symbols:

c1, c15, c16, c2

### (11) CdtKnowledgeProof (BOTH BOUNDS(ID, ID) transformation)

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

U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))

### (12) 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
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
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))
s(z0) → n__s(z0)
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
S tuples:

ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:

U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
Defined Rule Symbols:

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

Defined Pair Symbols:

U12', ACTIVATE, U11'

Compound Symbols:

c1, c15, c16, c2

### (13) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

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
snd(pair(z0, z1)) → z1
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))

### (14) Obligation:

Complexity Dependency Tuples Problem
Rules:

splitAt(0, z0) → pair(nil, z0)
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
s(z0) → n__s(z0)
Tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
S tuples:

ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
K tuples:

U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
Defined Rule Symbols:

splitAt, activate, natsFrom, s

Defined Pair Symbols:

U12', ACTIVATE, U11'

Compound Symbols:

c1, c15, c16, c2

### (15) CdtRuleRemovalProof (UPPER BOUND (ADD(O(n^1))) transformation)

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

ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
We considered the (Usable) Rules:

activate(z0) → z0
activate(n__s(z0)) → s(activate(z0))
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
s(z0) → n__s(z0)
And the Tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [2]
POL(ACTIVATE(x1)) = [3] + [2]x1
POL(U11'(x1, x2, x3, x4)) = [4]x1 + [5]x2 + [4]x3 + [2]x4
POL(U12'(x1, x2)) = [3] + [2]x2
POL(activate(x1)) = [2]x1
POL(c1(x1)) = x1
POL(c15(x1)) = x1
POL(c16(x1)) = x1
POL(c2(x1)) = x1
POL(cons(x1, x2)) = [2] + x1
POL(n__natsFrom(x1)) = [1] + x1
POL(n__s(x1)) = [4] + x1
POL(natsFrom(x1)) = [2] + x1
POL(nil) = [3]
POL(pair(x1, x2)) = [3]
POL(s(x1)) = [4] + x1
POL(splitAt(x1, x2)) = [2] + [2]x1 + [2]x2
POL(tt) = [4]

### (16) Obligation:

Complexity Dependency Tuples Problem
Rules:

splitAt(0, z0) → pair(nil, z0)
activate(n__natsFrom(z0)) → natsFrom(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
natsFrom(z0) → cons(z0, n__natsFrom(n__s(z0)))
natsFrom(z0) → n__natsFrom(z0)
s(z0) → n__s(z0)
Tuples:

U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
S tuples:none
K tuples:

U11'(tt, z0, z1, z2) → c2(U12'(splitAt(activate(z0), activate(z2)), activate(z1)))
U12'(pair(z0, z1), z2) → c1(ACTIVATE(z2))
ACTIVATE(n__natsFrom(z0)) → c15(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c16(ACTIVATE(z0))
Defined Rule Symbols:

splitAt, activate, natsFrom, s

Defined Pair Symbols:

U12', ACTIVATE, U11'

Compound Symbols:

c1, c15, c16, c2

### (17) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty