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

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

Strict Trs:
  { terms(N) -> cons(recip(sqr(N)))
  , sqr(0()) -> 0()
  , sqr(s()) -> s()
  , dbl(0()) -> 0()
  , dbl(s()) -> s()
  , add(0(), X) -> X
  , add(s(), Y) -> s()
  , first(0(), X) -> nil()
  , first(s(), cons(Y)) -> cons(Y) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

We add following weak dependency pairs:

Strict DPs:
  { terms^#(N) -> c_1(sqr^#(N))
  , sqr^#(0()) -> c_2()
  , sqr^#(s()) -> c_3()
  , dbl^#(0()) -> c_4()
  , dbl^#(s()) -> c_5()
  , add^#(0(), X) -> c_6()
  , add^#(s(), Y) -> c_7()
  , first^#(0(), X) -> c_8()
  , first^#(s(), cons(Y)) -> c_9() }

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:
  { terms^#(N) -> c_1(sqr^#(N))
  , sqr^#(0()) -> c_2()
  , sqr^#(s()) -> c_3()
  , dbl^#(0()) -> c_4()
  , dbl^#(s()) -> c_5()
  , add^#(0(), X) -> c_6()
  , add^#(s(), Y) -> c_7()
  , first^#(0(), X) -> c_8()
  , first^#(s(), cons(Y)) -> c_9() }
Strict Trs:
  { terms(N) -> cons(recip(sqr(N)))
  , sqr(0()) -> 0()
  , sqr(s()) -> s()
  , dbl(0()) -> 0()
  , dbl(s()) -> s()
  , add(0(), X) -> X
  , add(s(), Y) -> s()
  , first(0(), X) -> nil()
  , first(s(), cons(Y)) -> cons(Y) }
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:
  { terms^#(N) -> c_1(sqr^#(N))
  , sqr^#(0()) -> c_2()
  , sqr^#(s()) -> c_3()
  , dbl^#(0()) -> c_4()
  , dbl^#(s()) -> c_5()
  , add^#(0(), X) -> c_6()
  , add^#(s(), Y) -> c_7()
  , first^#(0(), X) -> c_8()
  , first^#(s(), cons(Y)) -> c_9() }
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:
  Uargs(c_1) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

         [cons](x1) = [2]                  
                                           
                [0] = [2]                  
                                           
                [s] = [2]                  
                                           
      [terms^#](x1) = [2] x1 + [2]         
                                           
          [c_1](x1) = [1] x1 + [1]         
                                           
        [sqr^#](x1) = [2] x1 + [2]         
                                           
              [c_2] = [1]                  
                                           
              [c_3] = [1]                  
                                           
        [dbl^#](x1) = [2] x1 + [2]         
                                           
              [c_4] = [1]                  
                                           
              [c_5] = [1]                  
                                           
    [add^#](x1, x2) = [2] x1 + [2]         
                                           
              [c_6] = [1]                  
                                           
              [c_7] = [1]                  
                                           
  [first^#](x1, x2) = [2] x1 + [1] x2 + [1]
                                           
              [c_8] = [0]                  
                                           
              [c_9] = [2]                  

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)).

Strict DPs: { terms^#(N) -> c_1(sqr^#(N)) }
Weak DPs:
  { sqr^#(0()) -> c_2()
  , sqr^#(s()) -> c_3()
  , dbl^#(0()) -> c_4()
  , dbl^#(s()) -> c_5()
  , add^#(0(), X) -> c_6()
  , add^#(s(), Y) -> c_7()
  , first^#(0(), X) -> c_8()
  , first^#(s(), cons(Y)) -> c_9() }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(1))

We estimate the number of application of {1} by applications of
Pre({1}) = {}. Here rules are labeled as follows:

  DPs:
    { 1: terms^#(N) -> c_1(sqr^#(N))
    , 2: sqr^#(0()) -> c_2()
    , 3: sqr^#(s()) -> c_3()
    , 4: dbl^#(0()) -> c_4()
    , 5: dbl^#(s()) -> c_5()
    , 6: add^#(0(), X) -> c_6()
    , 7: add^#(s(), Y) -> c_7()
    , 8: first^#(0(), X) -> c_8()
    , 9: first^#(s(), cons(Y)) -> c_9() }

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

Weak DPs:
  { terms^#(N) -> c_1(sqr^#(N))
  , sqr^#(0()) -> c_2()
  , sqr^#(s()) -> c_3()
  , dbl^#(0()) -> c_4()
  , dbl^#(s()) -> c_5()
  , add^#(0(), X) -> c_6()
  , add^#(s(), Y) -> c_7()
  , first^#(0(), X) -> c_8()
  , first^#(s(), cons(Y)) -> c_9() }
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.

{ terms^#(N) -> c_1(sqr^#(N))
, sqr^#(0()) -> c_2()
, sqr^#(s()) -> c_3()
, dbl^#(0()) -> c_4()
, dbl^#(s()) -> c_5()
, add^#(0(), X) -> c_6()
, add^#(s(), Y) -> c_7()
, first^#(0(), X) -> c_8()
, first^#(s(), cons(Y)) -> c_9() }

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))