* Step 1: DependencyPairs WORST_CASE(?,O(1))
+ Considered Problem:
- Strict TRS:
activate(X) -> X
activate(n__zeros()) -> zeros()
tail(cons(X,XS)) -> activate(XS)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{activate/1,tail/1,zeros/0} / {0/0,cons/2,n__zeros/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate,tail,zeros} and constructors {0,cons,n__zeros}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
activate#(X) -> c_1()
activate#(n__zeros()) -> c_2(zeros#())
tail#(cons(X,XS)) -> c_3(activate#(XS))
zeros#() -> c_4()
zeros#() -> c_5()
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__zeros()) -> c_2(zeros#())
tail#(cons(X,XS)) -> c_3(activate#(XS))
zeros#() -> c_4()
zeros#() -> c_5()
- Weak TRS:
activate(X) -> X
activate(n__zeros()) -> zeros()
tail(cons(X,XS)) -> activate(XS)
zeros() -> cons(0(),n__zeros())
zeros() -> n__zeros()
- Signature:
{activate/1,tail/1,zeros/0,activate#/1,tail#/1,zeros#/0} / {0/0,cons/2,n__zeros/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,tail#,zeros#} and constructors {0,cons
,n__zeros}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
activate#(X) -> c_1()
activate#(n__zeros()) -> c_2(zeros#())
tail#(cons(X,XS)) -> c_3(activate#(XS))
zeros#() -> c_4()
zeros#() -> c_5()
* Step 3: Trivial WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
activate#(X) -> c_1()
activate#(n__zeros()) -> c_2(zeros#())
tail#(cons(X,XS)) -> c_3(activate#(XS))
zeros#() -> c_4()
zeros#() -> c_5()
- Signature:
{activate/1,tail/1,zeros/0,activate#/1,tail#/1,zeros#/0} / {0/0,cons/2,n__zeros/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,tail#,zeros#} and constructors {0,cons
,n__zeros}
+ Applied Processor:
Trivial
+ Details:
Consider the dependency graph
1:S:activate#(X) -> c_1()

2:S:activate#(n__zeros()) -> c_2(zeros#())
-->_1 zeros#() -> c_5():5
-->_1 zeros#() -> c_4():4

3:S:tail#(cons(X,XS)) -> c_3(activate#(XS))
-->_1 activate#(n__zeros()) -> c_2(zeros#()):2
-->_1 activate#(X) -> c_1():1

4:S:zeros#() -> c_4()

5:S:zeros#() -> c_5()

The dependency graph contains no loops, we remove all dependency pairs.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{activate/1,tail/1,zeros/0,activate#/1,tail#/1,zeros#/0} / {0/0,cons/2,n__zeros/0,c_1/0,c_2/1,c_3/1,c_4/0
,c_5/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {activate#,tail#,zeros#} and constructors {0,cons
,n__zeros}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(1))