YES(O(1),O(1))

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

Strict Trs:
  { first(0(), X) -> nil()
  , first(s(X), cons(Y)) -> cons(Y)
  , from(X) -> cons(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add following weak dependency pairs:

Strict DPs:
  { first^#(0(), X) -> c_1()
  , first^#(s(X), cons(Y)) -> c_2()
  , from^#(X) -> c_3() }

and mark the set of starting terms.

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

Strict DPs:
  { first^#(0(), X) -> c_1()
  , first^#(s(X), cons(Y)) -> c_2()
  , from^#(X) -> c_3() }
Strict Trs:
  { first(0(), X) -> nil()
  , first(s(X), cons(Y)) -> cons(Y)
  , from(X) -> cons(X) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(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(1)).

Strict DPs:
  { first^#(0(), X) -> c_1()
  , first^#(s(X), cons(Y)) -> c_2()
  , from^#(X) -> c_3() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

The following argument positions are usable:
  none

TcT has computed following constructor-restricted matrix
interpretation.

                [0] = [1]                  
                                           
            [s](x1) = [1]                  
                                           
         [cons](x1) = [1]                  
                                           
  [first^#](x1, x2) = [2] x1 + [1] x2 + [0]
                                           
              [c_1] = [1]                  
                                           
              [c_2] = [2]                  
                                           
       [from^#](x1) = [2]                  
                                           
              [c_3] = [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:
  { first^#(0(), X) -> c_1()
  , first^#(s(X), cons(Y)) -> c_2()
  , from^#(X) -> c_3() }
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^#(0(), X) -> c_1()
, first^#(s(X), cons(Y)) -> c_2()
, from^#(X) -> c_3() }

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(1))