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:
  { f(a, empty()) -> g(a, empty())
  , f(a, cons(x, k)) -> f(cons(x, a), k)
  , g(empty(), d) -> d
  , g(cons(x, k), d) -> g(k, cons(x, d)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { f^#(a, empty()) -> c_1(g^#(a, empty()))
  , f^#(a, cons(x, k)) -> c_2(f^#(cons(x, a), k))
  , g^#(empty(), d) -> c_3()
  , g^#(cons(x, k), d) -> c_4(g^#(k, cons(x, d))) }

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:
  { f^#(a, empty()) -> c_1(g^#(a, empty()))
  , f^#(a, cons(x, k)) -> c_2(f^#(cons(x, a), k))
  , g^#(empty(), d) -> c_3()
  , g^#(cons(x, k), d) -> c_4(g^#(k, cons(x, d))) }
Strict Trs:
  { f(a, empty()) -> g(a, empty())
  , f(a, cons(x, k)) -> f(cons(x, a), k)
  , g(empty(), d) -> d
  , g(cons(x, k), d) -> g(k, cons(x, d)) }
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:
  { f^#(a, empty()) -> c_1(g^#(a, empty()))
  , f^#(a, cons(x, k)) -> c_2(f^#(cons(x, a), k))
  , g^#(empty(), d) -> c_3()
  , g^#(cons(x, k), d) -> c_4(g^#(k, cons(x, d))) }
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_2) = {1}, Uargs(c_4) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

         [empty] = [0]                  
                                        
  [cons](x1, x2) = [1] x1 + [1] x2 + [0]
                                        
   [f^#](x1, x2) = [0]                  
                                        
       [c_1](x1) = [1] x1 + [0]         
                                        
   [g^#](x1, x2) = [1]                  
                                        
       [c_2](x1) = [1] x1 + [1]         
                                        
           [c_3] = [0]                  
                                        
       [c_4](x1) = [1] x1 + [0]         

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:
  { f^#(a, empty()) -> c_1(g^#(a, empty()))
  , f^#(a, cons(x, k)) -> c_2(f^#(cons(x, a), k))
  , g^#(cons(x, k), d) -> c_4(g^#(k, cons(x, d))) }
Weak DPs: { g^#(empty(), d) -> c_3() }
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.

{ g^#(empty(), d) -> c_3() }

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

Strict DPs:
  { f^#(a, empty()) -> c_1(g^#(a, empty()))
  , f^#(a, cons(x, k)) -> c_2(f^#(cons(x, a), k))
  , g^#(cons(x, k), d) -> c_4(g^#(k, cons(x, d))) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The following argument positions are usable:
  Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_4) = {1}

TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).

         [empty] = [0]                  
                                        
  [cons](x1, x2) = [1] x1 + [1] x2 + [1]
                                        
   [f^#](x1, x2) = [2] x1 + [3] x2 + [1]
                                        
       [c_1](x1) = [2] x1 + [0]         
                                        
   [g^#](x1, x2) = [1] x1 + [0]         
                                        
       [c_2](x1) = [1] x1 + [0]         
                                        
       [c_4](x1) = [1] x1 + [0]         

This order satisfies following ordering constraints:

     [f^#(a, empty())] = [2] a + [1]                
                       > [2] a + [0]                
                       = [c_1(g^#(a, empty()))]     
                                                    
  [f^#(a, cons(x, k))] = [2] a + [3] x + [3] k + [4]
                       > [2] a + [2] x + [3] k + [3]
                       = [c_2(f^#(cons(x, a), k))]  
                                                    
  [g^#(cons(x, k), d)] = [1] x + [1] k + [1]        
                       > [1] k + [0]                
                       = [c_4(g^#(k, cons(x, d)))]  
                                                    

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