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:
  { @([](), xs) -> xs
  , @(::(x, xs), ys) -> ::(x, @(xs, ys))
  , flatten([]()) -> []()
  , flatten(::(x, xs)) -> @(x, flatten(xs)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following dependency tuples:

Strict DPs:
  { @^#([](), xs) -> c_1()
  , @^#(::(x, xs), ys) -> c_2(@^#(xs, ys))
  , flatten^#([]()) -> c_3()
  , flatten^#(::(x, xs)) -> c_4(@^#(x, flatten(xs)), flatten^#(xs)) }

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:
  { @^#([](), xs) -> c_1()
  , @^#(::(x, xs), ys) -> c_2(@^#(xs, ys))
  , flatten^#([]()) -> c_3()
  , flatten^#(::(x, xs)) -> c_4(@^#(x, flatten(xs)), flatten^#(xs)) }
Weak Trs:
  { @([](), xs) -> xs
  , @(::(x, xs), ys) -> ::(x, @(xs, ys))
  , flatten([]()) -> []()
  , flatten(::(x, xs)) -> @(x, flatten(xs)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We estimate the number of application of {1,3} by applications of
Pre({1,3}) = {2,4}. Here rules are labeled as follows:

  DPs:
    { 1: @^#([](), xs) -> c_1()
    , 2: @^#(::(x, xs), ys) -> c_2(@^#(xs, ys))
    , 3: flatten^#([]()) -> c_3()
    , 4: flatten^#(::(x, xs)) ->
         c_4(@^#(x, flatten(xs)), flatten^#(xs)) }

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

Strict DPs:
  { @^#(::(x, xs), ys) -> c_2(@^#(xs, ys))
  , flatten^#(::(x, xs)) -> c_4(@^#(x, flatten(xs)), flatten^#(xs)) }
Weak DPs:
  { @^#([](), xs) -> c_1()
  , flatten^#([]()) -> c_3() }
Weak Trs:
  { @([](), xs) -> xs
  , @(::(x, xs), ys) -> ::(x, @(xs, ys))
  , flatten([]()) -> []()
  , flatten(::(x, xs)) -> @(x, flatten(xs)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

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

{ @^#([](), xs) -> c_1()
, flatten^#([]()) -> c_3() }

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

Strict DPs:
  { @^#(::(x, xs), ys) -> c_2(@^#(xs, ys))
  , flatten^#(::(x, xs)) -> c_4(@^#(x, flatten(xs)), flatten^#(xs)) }
Weak Trs:
  { @([](), xs) -> xs
  , @(::(x, xs), ys) -> ::(x, @(xs, ys))
  , flatten([]()) -> []()
  , flatten(::(x, xs)) -> @(x, flatten(xs)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.

DPs:
  { 1: @^#(::(x, xs), ys) -> c_2(@^#(xs, ys))
  , 2: flatten^#(::(x, xs)) ->
       c_4(@^#(x, flatten(xs)), flatten^#(xs)) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_2) = {1}, Uargs(c_4) = {1, 2}
  
  TcT has computed following constructor-based matrix interpretation
  satisfying not(EDA).
  
               [[]] = [1]
                         
        [@](x1, x2) = [1] x1 + [1]
                                  
       [::](x1, x2) = [1] x1 + [1] x2 + [1]
                                           
      [flatten](x1) = [1]
                         
      [@^#](x1, x2) = [1] x1 + [0]
                                  
          [c_2](x1) = [1] x1 + [0]
                                  
    [flatten^#](x1) = [1] x1 + [1]
                                  
      [c_4](x1, x2) = [1] x1 + [1] x2 + [0]
  
  This order satisfies following ordering constraints
  
      [@^#(::(x, xs), ys)] = [1] x + [1] xs + [1]                     
                           > [1] xs + [0]                             
                           = [c_2(@^#(xs, ys))]                       
                                                                      
    [flatten^#(::(x, xs))] = [1] x + [1] xs + [2]                     
                           > [1] x + [1] xs + [1]                     
                           = [c_4(@^#(x, flatten(xs)), flatten^#(xs))]
                                                                      

The strictly oriented rules are moved into the corresponding weak
component(s).

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

Weak DPs:
  { @^#(::(x, xs), ys) -> c_2(@^#(xs, ys))
  , flatten^#(::(x, xs)) -> c_4(@^#(x, flatten(xs)), flatten^#(xs)) }
Weak Trs:
  { @([](), xs) -> xs
  , @(::(x, xs), ys) -> ::(x, @(xs, ys))
  , flatten([]()) -> []()
  , flatten(::(x, xs)) -> @(x, flatten(xs)) }
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.

{ @^#(::(x, xs), ys) -> c_2(@^#(xs, ys))
, flatten^#(::(x, xs)) -> c_4(@^#(x, flatten(xs)), flatten^#(xs)) }

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

Weak Trs:
  { @([](), xs) -> xs
  , @(::(x, xs), ys) -> ::(x, @(xs, ys))
  , flatten([]()) -> []()
  , flatten(::(x, xs)) -> @(x, flatten(xs)) }
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)).

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

Empty rules are trivially bounded

Wall-time: 0.349067s
CPU-time: 3.81842s

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