### (0) Obligation:

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

natsFrom(N) → cons(N, n__natsFrom(s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
tail(cons(N, XS)) → activate(XS)
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(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 Cpx (relative) TRS to CDT

### (2) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
fst(pair(z0, z1)) → z0
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2))
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1)
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
afterNth(z0, z1) → snd(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
Tuples:

NATSFROM(z0) → c
NATSFROM(z0) → c1
FST(pair(z0, z1)) → c2
SND(pair(z0, z1)) → c3
SPLITAT(0, z0) → c4
SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2))
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3))
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1))
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
ACTIVATE(n__natsFrom(z0)) → c12(NATSFROM(z0))
ACTIVATE(z0) → c13
S tuples:

NATSFROM(z0) → c
NATSFROM(z0) → c1
FST(pair(z0, z1)) → c2
SND(pair(z0, z1)) → c3
SPLITAT(0, z0) → c4
SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2))
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3))
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1))
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
ACTIVATE(n__natsFrom(z0)) → c12(NATSFROM(z0))
ACTIVATE(z0) → c13
K tuples:none
Defined Rule Symbols:

natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, activate

Defined Pair Symbols:

NATSFROM, FST, SND, SPLITAT, U, HEAD, TAIL, SEL, TAKE, AFTERNTH, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13

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

Removed 10 trailing nodes:

NATSFROM(z0) → c
NATSFROM(z0) → c1
ACTIVATE(n__natsFrom(z0)) → c12(NATSFROM(z0))
FST(pair(z0, z1)) → c2
SND(pair(z0, z1)) → c3
SPLITAT(0, z0) → c4
U(pair(z0, z1), z2, z3, z4) → c6(ACTIVATE(z3))
ACTIVATE(z0) → c13
TAIL(cons(z0, z1)) → c8(ACTIVATE(z1))

### (4) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
fst(pair(z0, z1)) → z0
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2))
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1)
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
afterNth(z0, z1) → snd(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
Tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2))
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
S tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(U(splitAt(z0, activate(z2)), z0, z1, activate(z2)), SPLITAT(z0, activate(z2)), ACTIVATE(z2), ACTIVATE(z2))
SEL(z0, z1) → c9(HEAD(afterNth(z0, z1)), AFTERNTH(z0, z1))
TAKE(z0, z1) → c10(FST(splitAt(z0, z1)), SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c11(SND(splitAt(z0, z1)), SPLITAT(z0, z1))
K tuples:none
Defined Rule Symbols:

natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, activate

Defined Pair Symbols:

SPLITAT, SEL, TAKE, AFTERNTH

Compound Symbols:

c5, c9, c10, c11

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

Removed 6 trailing tuple parts

### (6) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
fst(pair(z0, z1)) → z0
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2))
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1)
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
afterNth(z0, z1) → snd(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
Tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
SEL(z0, z1) → c9(AFTERNTH(z0, z1))
TAKE(z0, z1) → c10(SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c11(SPLITAT(z0, z1))
S tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
SEL(z0, z1) → c9(AFTERNTH(z0, z1))
TAKE(z0, z1) → c10(SPLITAT(z0, z1))
AFTERNTH(z0, z1) → c11(SPLITAT(z0, z1))
K tuples:none
Defined Rule Symbols:

natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, activate

Defined Pair Symbols:

SPLITAT, SEL, TAKE, AFTERNTH

Compound Symbols:

c5, c9, c10, c11

### (7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

TAKE(z0, z1) → c10(SPLITAT(z0, z1))
SEL(z0, z1) → c9(AFTERNTH(z0, z1))
AFTERNTH(z0, z1) → c11(SPLITAT(z0, z1))

### (8) Obligation:

Complexity Dependency Tuples Problem
Rules:

natsFrom(z0) → cons(z0, n__natsFrom(s(z0)))
natsFrom(z0) → n__natsFrom(z0)
fst(pair(z0, z1)) → z0
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2))
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1)
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
afterNth(z0, z1) → snd(splitAt(z0, z1))
activate(n__natsFrom(z0)) → natsFrom(z0)
activate(z0) → z0
Tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
S tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
K tuples:none
Defined Rule Symbols:

natsFrom, fst, snd, splitAt, u, head, tail, sel, take, afterNth, activate

Defined Pair Symbols:

SPLITAT

Compound Symbols:

c5

### (9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

fst(pair(z0, z1)) → z0
snd(pair(z0, z1)) → z1
splitAt(0, z0) → pair(nil, z0)
splitAt(s(z0), cons(z1, z2)) → u(splitAt(z0, activate(z2)), z0, z1, activate(z2))
u(pair(z0, z1), z2, z3, z4) → pair(cons(activate(z3), z0), z1)
tail(cons(z0, z1)) → activate(z1)
take(z0, z1) → fst(splitAt(z0, z1))
afterNth(z0, z1) → snd(splitAt(z0, z1))

### (10) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
S tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
K tuples:none
Defined Rule Symbols:

activate, natsFrom

Defined Pair Symbols:

SPLITAT

Compound Symbols:

c5

### (11) 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.

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
We considered the (Usable) Rules:none
And the Tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(SPLITAT(x1, x2)) = x1
POL(activate(x1)) = 0
POL(c5(x1)) = x1
POL(cons(x1, x2)) = 0
POL(n__natsFrom(x1)) = 0
POL(natsFrom(x1)) = [2] + [3]x1
POL(s(x1)) = [1] + x1

### (12) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
S tuples:none
K tuples:

SPLITAT(s(z0), cons(z1, z2)) → c5(SPLITAT(z0, activate(z2)))
Defined Rule Symbols:

activate, natsFrom

Defined Pair Symbols:

SPLITAT

Compound Symbols:

c5

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

The set S is empty