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

(0) Obligation:

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

2nd(cons(X, n__cons(Y, Z))) → activate(Y)
from(X) → cons(X, n__from(s(X)))
cons(X1, X2) → n__cons(X1, X2)
from(X) → n__from(X)
activate(n__cons(X1, X2)) → cons(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:

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

Tuples:

2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
FROM(z0) → c1(CONS(z0, n__from(s(z0))))
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
ACTIVATE(n__from(z0)) → c5(FROM(z0))

S tuples:

2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))
FROM(z0) → c1(CONS(z0, n__from(s(z0))))
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
ACTIVATE(n__from(z0)) → c5(FROM(z0))

K tuples:none
Defined Rule Symbols:

2nd, from, cons, activate

Defined Pair Symbols:

2ND, FROM, ACTIVATE

Compound Symbols:

c, c1, c4, c5

(3) CdtUnreachableProof (EQUIVALENT transformation)

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

2ND(cons(z0, n__cons(z1, z2))) → c(ACTIVATE(z1))

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:

FROM(z0) → c1(CONS(z0, n__from(s(z0))))
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
ACTIVATE(n__from(z0)) → c5(FROM(z0))

S tuples:

FROM(z0) → c1(CONS(z0, n__from(s(z0))))
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
ACTIVATE(n__from(z0)) → c5(FROM(z0))

K tuples:none
Defined Rule Symbols:

2nd, from, cons, activate

Defined Pair Symbols:

FROM, ACTIVATE

Compound Symbols:

c1, c4, c5

(5) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 3 of 3 dangling nodes:

FROM(z0) → c1(CONS(z0, n__from(s(z0))))
ACTIVATE(n__cons(z0, z1)) → c4(CONS(z0, z1))
ACTIVATE(n__from(z0)) → c5(FROM(z0))

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

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

Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:

2nd, from, cons, activate

Defined Pair Symbols:none

Compound Symbols:none

(7) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

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