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:

2nd(cons1(X, cons(Y, Z))) → Y
2nd(cons(X, X1)) → 2nd(cons1(X, activate(X1)))
from(X) → cons(X, n__from(n__s(X)))
from(X) → n__from(X)
s(X) → n__s(X)
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 Cpx (relative) TRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons1(z0, cons(z1, z2))) → z1
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1)))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0

Tuples:

2ND(cons1(z0, cons(z1, z2))) → c
2ND(cons(z0, z1)) → c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
FROM(z0) → c2
FROM(z0) → c3
S(z0) → c4
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c7

S tuples:

2ND(cons1(z0, cons(z1, z2))) → c
2ND(cons(z0, z1)) → c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
FROM(z0) → c2
FROM(z0) → c3
S(z0) → c4
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))
ACTIVATE(z0) → c7

K tuples:none
Defined Rule Symbols:

2nd, from, s, activate

Defined Pair Symbols:

2ND, FROM, S, ACTIVATE

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7

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

Removed 5 trailing nodes:

2ND(cons1(z0, cons(z1, z2))) → c
FROM(z0) → c2
FROM(z0) → c3
S(z0) → c4
ACTIVATE(z0) → c7

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons1(z0, cons(z1, z2))) → z1
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1)))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0

Tuples:

2ND(cons(z0, z1)) → c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))

S tuples:

2ND(cons(z0, z1)) → c1(2ND(cons1(z0, activate(z1))), ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c5(FROM(activate(z0)), ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(S(activate(z0)), ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

2nd, from, s, activate

Defined Pair Symbols:

2ND, ACTIVATE

Compound Symbols:

c1, c5, c6

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

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons1(z0, cons(z1, z2))) → z1
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1)))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0

Tuples:

2ND(cons(z0, z1)) → c1(ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

S tuples:

2ND(cons(z0, z1)) → c1(ACTIVATE(z1))
ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

2nd, from, s, activate

Defined Pair Symbols:

2ND, ACTIVATE

Compound Symbols:

c1, c5, c6

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication transformation)

Removed 1 leading nodes:

2ND(cons(z0, z1)) → c1(ACTIVATE(z1))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

2nd(cons1(z0, cons(z1, z2))) → z1
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1)))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0

Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

S tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:

2nd, from, s, activate

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

(9) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

2nd(cons1(z0, cons(z1, z2))) → z1
2nd(cons(z0, z1)) → 2nd(cons1(z0, activate(z1)))
from(z0) → cons(z0, n__from(n__s(z0)))
from(z0) → n__from(z0)
s(z0) → n__s(z0)
activate(n__from(z0)) → from(activate(z0))
activate(n__s(z0)) → s(activate(z0))
activate(z0) → z0

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

S tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

(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.

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

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

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

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

POL(ACTIVATE(x₁)) = [4]x₁   
POL(c5(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(n__from(x₁)) = [1] + x₁   
POL(n__s(x₁)) = [4] + x₁   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:none
Tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

S tuples:none
K tuples:

ACTIVATE(n__from(z0)) → c5(ACTIVATE(z0))
ACTIVATE(n__s(z0)) → c6(ACTIVATE(z0))

Defined Rule Symbols:none

Defined Pair Symbols:

ACTIVATE

Compound Symbols:

c5, c6

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

The set S is empty

(14) BOUNDS(O(1), O(1))