YES(O(1),O(n^1))

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict Trs:
  { zeros() -> cons(0(), n__zeros())
  , zeros() -> n__zeros()
  , tail(cons(X, XS)) -> activate(XS)
  , activate(X) -> X
  , activate(n__zeros()) -> zeros() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

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

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { zeros^#() -> c_1()
  , zeros^#() -> c_2()
  , tail^#(cons(X, XS)) -> c_3(activate^#(XS))
  , activate^#(X) -> c_4()
  , activate^#(n__zeros()) -> c_5(zeros^#()) }
Strict Trs:
  { zeros() -> cons(0(), n__zeros())
  , zeros() -> n__zeros()
  , tail(cons(X, XS)) -> activate(XS)
  , activate(X) -> X
  , activate(n__zeros()) -> zeros() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

No rule is usable, rules are removed from the input problem.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).

Strict DPs:
  { zeros^#() -> c_1()
  , zeros^#() -> c_2()
  , tail^#(cons(X, XS)) -> c_3(activate^#(XS))
  , activate^#(X) -> c_4()
  , activate^#(n__zeros()) -> c_5(zeros^#()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(c_3) = {1}, Uargs(c_5) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

    [cons](x1, x2) = [1] x2 + [2]
                                 
        [n__zeros] = [1]         
                                 
         [zeros^#] = [1]         
                                 
             [c_1] = [0]         
                                 
             [c_2] = [0]         
                                 
      [tail^#](x1) = [2] x1 + [2]
                                 
         [c_3](x1) = [1] x1 + [1]
                                 
  [activate^#](x1) = [1] x1 + [2]
                                 
             [c_4] = [1]         
                                 
         [c_5](x1) = [1] x1 + [1]

This order satisfies following ordering constraints:


Further, it can be verified that all rules not oriented are covered by the weightgap condition.

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Weak DPs:
  { zeros^#() -> c_1()
  , zeros^#() -> c_2()
  , tail^#(cons(X, XS)) -> c_3(activate^#(XS))
  , activate^#(X) -> c_4()
  , activate^#(n__zeros()) -> c_5(zeros^#()) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.

{ zeros^#() -> c_1()
, zeros^#() -> c_2()
, tail^#(cons(X, XS)) -> c_3(activate^#(XS))
, activate^#(X) -> c_4()
, activate^#(n__zeros()) -> c_5(zeros^#()) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(1)).

Rules: Empty
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

The input was oriented with the instance of 'Small Polynomial Path
Order (PS)' as induced by the safe mapping


and precedence

 empty .

Following symbols are considered recursive:

 {}

The recursion depth is 0.

Further, following argument filtering is employed:

 empty

Usable defined function symbols are a subset of:

 {}

For your convenience, here are the satisfied ordering constraints:


Hurray, we answered YES(O(1),O(n^1))