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:
  { first(X1, X2) -> n__first(X1, X2)
  , first(0(), X) -> nil()
  , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
  , activate(X) -> X
  , activate(n__first(X1, X2)) -> first(X1, X2)
  , activate(n__from(X)) -> from(X)
  , from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { first^#(X1, X2) -> c_1()
  , first^#(0(), X) -> c_2()
  , first^#(s(X), cons(Y, Z)) -> c_3(activate^#(Z))
  , activate^#(X) -> c_4()
  , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
  , activate^#(n__from(X)) -> c_6(from^#(X))
  , from^#(X) -> c_7()
  , from^#(X) -> c_8() }

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:
  { first^#(X1, X2) -> c_1()
  , first^#(0(), X) -> c_2()
  , first^#(s(X), cons(Y, Z)) -> c_3(activate^#(Z))
  , activate^#(X) -> c_4()
  , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
  , activate^#(n__from(X)) -> c_6(from^#(X))
  , from^#(X) -> c_7()
  , from^#(X) -> c_8() }
Strict Trs:
  { first(X1, X2) -> n__first(X1, X2)
  , first(0(), X) -> nil()
  , first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
  , activate(X) -> X
  , activate(n__first(X1, X2)) -> first(X1, X2)
  , activate(n__from(X)) -> from(X)
  , from(X) -> cons(X, n__from(s(X)))
  , from(X) -> n__from(X) }
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:
  { first^#(X1, X2) -> c_1()
  , first^#(0(), X) -> c_2()
  , first^#(s(X), cons(Y, Z)) -> c_3(activate^#(Z))
  , activate^#(X) -> c_4()
  , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
  , activate^#(n__from(X)) -> c_6(from^#(X))
  , from^#(X) -> c_7()
  , from^#(X) -> c_8() }
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}, Uargs(c_6) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

                 [0] = [1]                  
                                            
             [s](x1) = [1]                  
                                            
      [cons](x1, x2) = [1] x2 + [1]         
                                            
  [n__first](x1, x2) = [1] x1 + [1] x2 + [2]
                                            
       [n__from](x1) = [1] x1 + [1]         
                                            
   [first^#](x1, x2) = [1] x1 + [2] x2 + [1]
                                            
               [c_1] = [0]                  
                                            
               [c_2] = [1]                  
                                            
           [c_3](x1) = [1] x1 + [1]         
                                            
    [activate^#](x1) = [2] x1 + [2]         
                                            
               [c_4] = [1]                  
                                            
           [c_5](x1) = [1] x1 + [1]         
                                            
           [c_6](x1) = [1] x1 + [1]         
                                            
        [from^#](x1) = [2] x1 + [1]         
                                            
               [c_7] = [0]                  
                                            
               [c_8] = [0]                  

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:
  { first^#(X1, X2) -> c_1()
  , first^#(0(), X) -> c_2()
  , first^#(s(X), cons(Y, Z)) -> c_3(activate^#(Z))
  , activate^#(X) -> c_4()
  , activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
  , activate^#(n__from(X)) -> c_6(from^#(X))
  , from^#(X) -> c_7()
  , from^#(X) -> c_8() }
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.

{ first^#(X1, X2) -> c_1()
, first^#(0(), X) -> c_2()
, first^#(s(X), cons(Y, Z)) -> c_3(activate^#(Z))
, activate^#(X) -> c_4()
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
, activate^#(n__from(X)) -> c_6(from^#(X))
, from^#(X) -> c_7()
, from^#(X) -> c_8() }

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))