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:

from(X) → cons(X, n__from(s(X)))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, activate(Z))
from(X) → n__from(X)
activate(n__from(X)) → from(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:

from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
activate(n__from(z0)) → from(z0)
activate(z0) → z0

Tuples:

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c4(FROM(z0))

S tuples:

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)), ACTIVATE(z2))
ACTIVATE(n__from(z0)) → c4(FROM(z0))

K tuples:none
Defined Rule Symbols:

from, sel, activate

Defined Pair Symbols:

SEL, ACTIVATE

Compound Symbols:

c3, c4

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 2 dangling nodes:

ACTIVATE(n__from(z0)) → c4(FROM(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
activate(n__from(z0)) → from(z0)
activate(z0) → z0

Tuples:

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)), ACTIVATE(z2))

S tuples:

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)), ACTIVATE(z2))

K tuples:none
Defined Rule Symbols:

from, sel, activate

Defined Pair Symbols:

SEL

Compound Symbols:

c3

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

Removed 1 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
activate(n__from(z0)) → from(z0)
activate(z0) → z0

Tuples:

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)))

S tuples:

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)))

K tuples:none
Defined Rule Symbols:

from, sel, activate

Defined Pair Symbols:

SEL

Compound Symbols:

c3

(7) 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.

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)))

We considered the (Usable) Rules:

activate(n__from(z0)) → from(z0)
activate(z0) → z0
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)

And the Tuples:

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)))

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

POL(SEL(x₁, x₂)) = x₁   
POL(activate(x₁)) = 0   
POL(c3(x₁)) = x₁   
POL(cons(x₁, x₂)) = 0   
POL(from(x₁)) = [2] + [3]x₁   
POL(n__from(x₁)) = 0   
POL(s(x₁)) = [1] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
sel(0, cons(z0, z1)) → z0
sel(s(z0), cons(z1, z2)) → sel(z0, activate(z2))
activate(n__from(z0)) → from(z0)
activate(z0) → z0

Tuples:

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)))

S tuples:none
K tuples:

SEL(s(z0), cons(z1, z2)) → c3(SEL(z0, activate(z2)))

Defined Rule Symbols:

from, sel, activate

Defined Pair Symbols:

SEL

Compound Symbols:

c3

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(10) BOUNDS(O(1), O(1))