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

0 CpxTRS
↳1 CpxTrsToCdtProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CdtProblem
↳3 CdtUnreachableProof (⇔, 0 ms)
↳4 CdtProblem
↳5 CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID), 0 ms)
↳6 CdtProblem
↳7 CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))), 98 ms)
↳8 CdtProblem
↳9 SIsEmptyProof (⇔, 0 ms)
↳10 BOUNDS(O(1), O(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(n__s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
s(X) → n__s(X)
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
activate(n__from(X)) → from(activate(X))
activate(n__s(X)) → s(activate(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(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c6(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(S(activate(z0)), ACTIVATE(z0))
S tuples:

FIRST(s(z0), cons(z1, z2)) → c1(ACTIVATE(z2))
ACTIVATE(n__first(z0, z1)) → c6(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

first, from, s, activate

Defined Pair Symbols:

FIRST, ACTIVATE

Compound Symbols:

c1, c6, c7, c8

(3) CdtUnreachableProof (EQUIVALENT transformation)

The following tuples could be removed as they are not reachable from basic start terms:

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

(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(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__first(z0, z1)) → c6(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(S(activate(z0)), ACTIVATE(z0))
S tuples:

ACTIVATE(n__first(z0, z1)) → c6(FIRST(activate(z0), activate(z1)), ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(S(activate(z0)), ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

first, from, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

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

Removed 3 trailing tuple parts

(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(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
S tuples:

ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:

first, from, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

(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.

ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
We considered the (Usable) Rules:none
And the Tuples:

ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :

POL(ACTIVATE(x1)) = x1   
POL(c6(x1, x2)) = x1 + x2   
POL(c7(x1)) = x1   
POL(c8(x1)) = x1   
POL(n__first(x1, x2)) = [1] + x1 + x2   
POL(n__from(x1)) = [1] + x1   
POL(n__s(x1)) = [1] + x1   

(8) 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(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
activate(n__first(z0, z1)) → first(activate(z0), activate(z1))
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0
Tuples:

ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
S tuples:none
K tuples:

ACTIVATE(n__first(z0, z1)) → c6(ACTIVATE(z0), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c7(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c8(ACTIVATE(z0))
Defined Rule Symbols:

first, from, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c6, c7, c8

(9) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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