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:

zeros → cons(0, n__zeros)
tail(cons(X, XS)) → activate(XS)
zeros → n__zeros
activate(n__zeros) → zeros
activate(X) → X

Rewrite Strategy: INNERMOST

(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)

Converted CpxTRS to CDT

(2) Obligation:

Complexity Dependency Tuples Problem
Rules:

zeros → cons(0, n__zeros)
zeros → n__zeros
tail(cons(z0, z1)) → activate(z1)
activate(n__zeros) → zeros
activate(z0) → z0

Tuples:

TAIL(cons(z0, z1)) → c2(ACTIVATE(z1))
ACTIVATE(n__zeros) → c3(ZEROS)

S tuples:

TAIL(cons(z0, z1)) → c2(ACTIVATE(z1))
ACTIVATE(n__zeros) → c3(ZEROS)

K tuples:none
Defined Rule Symbols:

zeros, tail, activate

Defined Pair Symbols:

TAIL, ACTIVATE

Compound Symbols:

c2, c3

(3) CdtGraphRemoveDanglingProof (ComplexityIfPolyImplication transformation)

Removed 2 of 2 dangling nodes:

TAIL(cons(z0, z1)) → c2(ACTIVATE(z1))
ACTIVATE(n__zeros) → c3(ZEROS)

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

zeros → cons(0, n__zeros)
zeros → n__zeros
tail(cons(z0, z1)) → activate(z1)
activate(n__zeros) → zeros
activate(z0) → z0

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

zeros, tail, activate

Defined Pair Symbols:none

Compound Symbols:none

(5) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(6) BOUNDS(O(1), O(1))