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:

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
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:

first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(n__from(z0)) → from(z0)
activate(z0) → z0

Tuples:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(z0, z1))
ACTIVATE(n__from(z0)) → c6(FROM(z0))

S tuples:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(z0, z1))
ACTIVATE(n__from(z0)) → c6(FROM(z0))

K tuples:none
Defined Rule Symbols:

first, from, activate

Defined Pair Symbols:

FIRST, ACTIVATE

Compound Symbols:

c1, c5, c6

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 1 of 3 dangling nodes:

ACTIVATE(n__from(z0)) → c6(FROM(z0))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(n__from(z0)) → from(z0)
activate(z0) → z0

Tuples:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(z0, z1))

S tuples:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(z0, z1))

K tuples:none
Defined Rule Symbols:

first, from, activate

Defined Pair Symbols:

FIRST, ACTIVATE

Compound Symbols:

c1, c5

(5) 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.

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(z0, z1))

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

POL(ACTIVATE(x₁)) = [4] + [4]x₁   
POL(FIRST(x₁, x₂)) = [2] + [4]x₂   
POL(c1(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(cons(x₁, x₂)) = [5] + x₂   
POL(n__first(x₁, x₂)) = [5] + x₂   
POL(s(x₁)) = [3]   

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

first(0, z0) → nil
first(s(z0), cons(z1, z2)) → cons(z1, n__first(z0, activate(z2)))
first(z0, z1) → n__first(z0, z1)
from(z0) → cons(z0, n__from(s(z0)))
from(z0) → n__from(z0)
activate(n__first(z0, z1)) → first(z0, z1)
activate(n__from(z0)) → from(z0)
activate(z0) → z0

Tuples:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(z0, z1))

S tuples:none
K tuples:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c5(FIRST(z0, z1))

Defined Rule Symbols:

first, from, activate

Defined Pair Symbols:

FIRST, ACTIVATE

Compound Symbols:

c1, c5

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(8) BOUNDS(O(1), O(1))