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(Cons(x', xs'), Cons(x, xs)) ->
    merge[Ite][False][Ite][False][Ite](<=(x', x),
                                       Cons(x', xs'),
                                       Cons(x, xs))
  , merge(Cons(x, xs), Nil()) -> Cons(x, xs)
  , merge(Nil(), ys) -> ys
  , goal(xs, ys) -> merge(xs, ys) }
Weak Trs:
  { <=(S(x), S(y)) -> <=(x, y)
  , <=(S(x), 0()) -> False()
  , <=(0(), y) -> True() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following innermost weak dependency pairs:

Strict DPs:
  { merge^#(Cons(x', xs'), Cons(x, xs)) -> c_1(<=^#(x', x))
  , merge^#(Cons(x, xs), Nil()) -> c_2()
  , merge^#(Nil(), ys) -> c_3()
  , goal^#(xs, ys) -> c_4(merge^#(xs, ys)) }
Weak DPs:
  { <=^#(S(x), S(y)) -> c_5(<=^#(x, y))
  , <=^#(S(x), 0()) -> c_6()
  , <=^#(0(), y) -> c_7() }

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^#(Cons(x', xs'), Cons(x, xs)) -> c_1(<=^#(x', x))
  , merge^#(Cons(x, xs), Nil()) -> c_2()
  , merge^#(Nil(), ys) -> c_3()
  , goal^#(xs, ys) -> c_4(merge^#(xs, ys)) }
Strict Trs:
  { merge(Cons(x', xs'), Cons(x, xs)) ->
    merge[Ite][False][Ite][False][Ite](<=(x', x),
                                       Cons(x', xs'),
                                       Cons(x, xs))
  , merge(Cons(x, xs), Nil()) -> Cons(x, xs)
  , merge(Nil(), ys) -> ys
  , goal(xs, ys) -> merge(xs, ys) }
Weak DPs:
  { <=^#(S(x), S(y)) -> c_5(<=^#(x, y))
  , <=^#(S(x), 0()) -> c_6()
  , <=^#(0(), y) -> c_7() }
Weak Trs:
  { <=(S(x), S(y)) -> <=(x, y)
  , <=(S(x), 0()) -> False()
  , <=(0(), y) -> True() }
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^#(Cons(x', xs'), Cons(x, xs)) -> c_1(<=^#(x', x))
  , merge^#(Cons(x, xs), Nil()) -> c_2()
  , merge^#(Nil(), ys) -> c_3()
  , goal^#(xs, ys) -> c_4(merge^#(xs, ys)) }
Weak DPs:
  { <=^#(S(x), S(y)) -> c_5(<=^#(x, y))
  , <=^#(S(x), 0()) -> c_6()
  , <=^#(0(), y) -> c_7() }
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_1) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1}

TcT has computed the following constructor-restricted matrix
interpretation.

            [S](x1) = [1 0] x1 + [0]           
                      [0 0]      [0]           
                                               
     [Cons](x1, x2) = [1 0] x1 + [0]           
                      [0 0]      [0]           
                                               
              [Nil] = [0]                      
                      [0]                      
                                               
                [0] = [0]                      
                      [0]                      
                                               
  [merge^#](x1, x2) = [2]                      
                      [0]                      
                                               
          [c_1](x1) = [1 0] x1 + [0]           
                      [0 1]      [0]           
                                               
  [<=^#](x1, x2) = [1]                      
                      [0]                      
                                               
              [c_2] = [1]                      
                      [0]                      
                                               
              [c_3] = [1]                      
                      [0]                      
                                               
   [goal^#](x1, x2) = [2 1] x1 + [1 2] x2 + [2]
                      [2 1]      [2 1]      [2]
                                               
          [c_4](x1) = [1 0] x1 + [1]           
                      [0 1]      [2]           
                                               
          [c_5](x1) = [1 0] x1 + [0]           
                      [0 1]      [0]           
                                               
              [c_6] = [1]                      
                      [0]                      
                                               
              [c_7] = [1]                      
                      [0]                      

The following symbols are considered usable

  {merge^#, <=^#, goal^#}

The order satisfies the following ordering constraints:

  [merge^#(Cons(x', xs'), Cons(x, xs))] =  [2]                      
                                           [0]                      
                                        >  [1]                      
                                           [0]                      
                                        =  [c_1(<=^#(x', x))]    
                                                                    
          [merge^#(Cons(x, xs), Nil())] =  [2]                      
                                           [0]                      
                                        >  [1]                      
                                           [0]                      
                                        =  [c_2()]                  
                                                                    
                   [merge^#(Nil(), ys)] =  [2]                      
                                           [0]                      
                                        >  [1]                      
                                           [0]                      
                                        =  [c_3()]                  
                                                                    
                  [<=^#(S(x), S(y))] =  [1]                      
                                           [0]                      
                                        >= [1]                      
                                           [0]                      
                                        =  [c_5(<=^#(x, y))]     
                                                                    
                   [<=^#(S(x), 0())] =  [1]                      
                                           [0]                      
                                        >= [1]                      
                                           [0]                      
                                        =  [c_6()]                  
                                                                    
                      [<=^#(0(), y)] =  [1]                      
                                           [0]                      
                                        >= [1]                      
                                           [0]                      
                                        =  [c_7()]                  
                                                                    
                       [goal^#(xs, ys)] =  [2 1] xs + [1 2] ys + [2]
                                           [2 1]      [2 1]      [2]
                                        ?  [3]                      
                                           [2]                      
                                        =  [c_4(merge^#(xs, ys))]   
                                                                    

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),O(1)).

Strict DPs: { goal^#(xs, ys) -> c_4(merge^#(xs, ys)) }
Weak DPs:
  { merge^#(Cons(x', xs'), Cons(x, xs)) -> c_1(<=^#(x', x))
  , merge^#(Cons(x, xs), Nil()) -> c_2()
  , merge^#(Nil(), ys) -> c_3()
  , <=^#(S(x), S(y)) -> c_5(<=^#(x, y))
  , <=^#(S(x), 0()) -> c_6()
  , <=^#(0(), y) -> c_7() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

  DPs:
    { 1: goal^#(xs, ys) -> c_4(merge^#(xs, ys))
    , 2: merge^#(Cons(x', xs'), Cons(x, xs)) -> c_1(<=^#(x', x))
    , 3: merge^#(Cons(x, xs), Nil()) -> c_2()
    , 4: merge^#(Nil(), ys) -> c_3()
    , 5: <=^#(S(x), S(y)) -> c_5(<=^#(x, y))
    , 6: <=^#(S(x), 0()) -> c_6()
    , 7: <=^#(0(), y) -> c_7() }

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

Weak DPs:
  { merge^#(Cons(x', xs'), Cons(x, xs)) -> c_1(<=^#(x', x))
  , merge^#(Cons(x, xs), Nil()) -> c_2()
  , merge^#(Nil(), ys) -> c_3()
  , <=^#(S(x), S(y)) -> c_5(<=^#(x, y))
  , <=^#(S(x), 0()) -> c_6()
  , <=^#(0(), y) -> c_7()
  , goal^#(xs, ys) -> c_4(merge^#(xs, ys)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

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

{ merge^#(Cons(x', xs'), Cons(x, xs)) -> c_1(<=^#(x', x))
, merge^#(Cons(x, xs), Nil()) -> c_2()
, merge^#(Nil(), ys) -> c_3()
, <=^#(S(x), S(y)) -> c_5(<=^#(x, y))
, <=^#(S(x), 0()) -> c_6()
, <=^#(0(), y) -> c_7()
, goal^#(xs, ys) -> c_4(merge^#(xs, ys)) }

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

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

Empty rules are trivially bounded

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