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:
  { merge(x, nil()) -> x
  , merge(nil(), y) -> y
  , merge(++(x, y), ++(u(), v())) -> ++(x, merge(y, ++(u(), v())))
  , merge(++(x, y), ++(u(), v())) -> ++(u(), merge(++(x, y), v())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { merge^#(x, nil()) -> c_1()
  , merge^#(nil(), y) -> c_2()
  , merge^#(++(x, y), ++(u(), v())) -> c_3(merge^#(y, ++(u(), v())))
  , merge^#(++(x, y), ++(u(), v())) -> c_4(merge^#(++(x, y), v())) }

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:
  { merge^#(x, nil()) -> c_1()
  , merge^#(nil(), y) -> c_2()
  , merge^#(++(x, y), ++(u(), v())) -> c_3(merge^#(y, ++(u(), v())))
  , merge^#(++(x, y), ++(u(), v())) -> c_4(merge^#(++(x, y), v())) }
Strict Trs:
  { merge(x, nil()) -> x
  , merge(nil(), y) -> y
  , merge(++(x, y), ++(u(), v())) -> ++(x, merge(y, ++(u(), v())))
  , merge(++(x, y), ++(u(), v())) -> ++(u(), merge(++(x, y), v())) }
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:
  { merge^#(x, nil()) -> c_1()
  , merge^#(nil(), y) -> c_2()
  , merge^#(++(x, y), ++(u(), v())) -> c_3(merge^#(y, ++(u(), v())))
  , merge^#(++(x, y), ++(u(), v())) -> c_4(merge^#(++(x, y), v())) }
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}

TcT has computed following constructor-restricted matrix
interpretation.

              [nil] = [0]         
                                  
       [++](x1, x2) = [2]         
                                  
                [u] = [0]         
                                  
                [v] = [0]         
                                  
  [merge^#](x1, x2) = [2] x2 + [0]
                                  
              [c_1] = [1]         
                                  
              [c_2] = [1]         
                                  
          [c_3](x1) = [1] x1 + [1]
                                  
          [c_4](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(n^1)).

Strict DPs:
  { merge^#(x, nil()) -> c_1()
  , merge^#(nil(), y) -> c_2()
  , merge^#(++(x, y), ++(u(), v())) ->
    c_3(merge^#(y, ++(u(), v()))) }
Weak DPs:
  { merge^#(++(x, y), ++(u(), v())) -> c_4(merge^#(++(x, y), v())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

We estimate the number of application of {1,2} by applications of
Pre({1,2}) = {3}. Here rules are labeled as follows:

  DPs:
    { 1: merge^#(x, nil()) -> c_1()
    , 2: merge^#(nil(), y) -> c_2()
    , 3: merge^#(++(x, y), ++(u(), v())) ->
         c_3(merge^#(y, ++(u(), v())))
    , 4: merge^#(++(x, y), ++(u(), v())) ->
         c_4(merge^#(++(x, y), v())) }

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

Strict DPs:
  { merge^#(++(x, y), ++(u(), v())) ->
    c_3(merge^#(y, ++(u(), v()))) }
Weak DPs:
  { merge^#(x, nil()) -> c_1()
  , merge^#(nil(), y) -> c_2()
  , merge^#(++(x, y), ++(u(), v())) -> c_4(merge^#(++(x, y), v())) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

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

{ merge^#(x, nil()) -> c_1()
, merge^#(nil(), y) -> c_2()
, merge^#(++(x, y), ++(u(), v())) -> c_4(merge^#(++(x, y), v())) }

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

Strict DPs:
  { merge^#(++(x, y), ++(u(), v())) ->
    c_3(merge^#(y, ++(u(), v()))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

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

 safe(++) = {1, 2}, safe(u) = {}, safe(v) = {}, safe(merge^#) = {2},
 safe(c_3) = {}

and precedence

 empty .

Following symbols are considered recursive:

 {merge^#}

The recursion depth is 1.

Further, following argument filtering is employed:

 pi(++) = [1, 2], pi(u) = [], pi(v) = [], pi(merge^#) = [1, 2],
 pi(c_3) = [1]

Usable defined function symbols are a subset of:

 {merge^#}

For your convenience, here are the satisfied ordering constraints:

  pi(merge^#(++(x, y), ++(u(), v()))) = merge^#(++(; x,  y); ++(; u(),  v()))
                                      > c_3(merge^#(y; ++(; u(),  v()));)    
                                      = pi(c_3(merge^#(y, ++(u(), v()))))    
                                                                             

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