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:
  { is_empty(nil()) -> true()
  , is_empty(cons(x, l)) -> false()
  , hd(cons(x, l)) -> x
  , tl(cons(x, l)) -> l
  , append(l1, l2) -> ifappend(l1, l2, l1)
  , ifappend(l1, l2, nil()) -> l2
  , ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { is_empty^#(nil()) -> c_1()
  , is_empty^#(cons(x, l)) -> c_2()
  , hd^#(cons(x, l)) -> c_3()
  , tl^#(cons(x, l)) -> c_4()
  , append^#(l1, l2) -> c_5(ifappend^#(l1, l2, l1))
  , ifappend^#(l1, l2, nil()) -> c_6()
  , ifappend^#(l1, l2, cons(x, l)) -> c_7(append^#(l, l2)) }

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:
  { is_empty^#(nil()) -> c_1()
  , is_empty^#(cons(x, l)) -> c_2()
  , hd^#(cons(x, l)) -> c_3()
  , tl^#(cons(x, l)) -> c_4()
  , append^#(l1, l2) -> c_5(ifappend^#(l1, l2, l1))
  , ifappend^#(l1, l2, nil()) -> c_6()
  , ifappend^#(l1, l2, cons(x, l)) -> c_7(append^#(l, l2)) }
Strict Trs:
  { is_empty(nil()) -> true()
  , is_empty(cons(x, l)) -> false()
  , hd(cons(x, l)) -> x
  , tl(cons(x, l)) -> l
  , append(l1, l2) -> ifappend(l1, l2, l1)
  , ifappend(l1, l2, nil()) -> l2
  , ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2)) }
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:
  { is_empty^#(nil()) -> c_1()
  , is_empty^#(cons(x, l)) -> c_2()
  , hd^#(cons(x, l)) -> c_3()
  , tl^#(cons(x, l)) -> c_4()
  , append^#(l1, l2) -> c_5(ifappend^#(l1, l2, l1))
  , ifappend^#(l1, l2, nil()) -> c_6()
  , ifappend^#(l1, l2, cons(x, l)) -> c_7(append^#(l, l2)) }
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_5) = {1}, Uargs(c_7) = {1}

TcT has computed following constructor-restricted matrix
interpretation.

                     [nil] = [2]                  
                                                  
            [cons](x1, x2) = [1] x2 + [1]         
                                                  
          [is_empty^#](x1) = [2] x1 + [2]         
                                                  
                     [c_1] = [1]                  
                                                  
                     [c_2] = [1]                  
                                                  
                [hd^#](x1) = [2] x1 + [1]         
                                                  
                     [c_3] = [2]                  
                                                  
                [tl^#](x1) = [2] x1 + [1]         
                                                  
                     [c_4] = [2]                  
                                                  
        [append^#](x1, x2) = [2] x1 + [2] x2 + [2]
                                                  
                 [c_5](x1) = [1] x1 + [1]         
                                                  
  [ifappend^#](x1, x2, x3) = [2] x2 + [2] x3 + [2]
                                                  
                     [c_6] = [1]                  
                                                  
                 [c_7](x1) = [1] 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: { append^#(l1, l2) -> c_5(ifappend^#(l1, l2, l1)) }
Weak DPs:
  { is_empty^#(nil()) -> c_1()
  , is_empty^#(cons(x, l)) -> c_2()
  , hd^#(cons(x, l)) -> c_3()
  , tl^#(cons(x, l)) -> c_4()
  , ifappend^#(l1, l2, nil()) -> c_6()
  , ifappend^#(l1, l2, cons(x, l)) -> c_7(append^#(l, l2)) }
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.

{ is_empty^#(nil()) -> c_1()
, is_empty^#(cons(x, l)) -> c_2()
, hd^#(cons(x, l)) -> c_3()
, tl^#(cons(x, l)) -> c_4()
, ifappend^#(l1, l2, nil()) -> c_6() }

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

Strict DPs: { append^#(l1, l2) -> c_5(ifappend^#(l1, l2, l1)) }
Weak DPs:
  { ifappend^#(l1, l2, cons(x, l)) -> c_7(append^#(l, l2)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The following argument positions are usable:
  Uargs(c_5) = {1}, Uargs(c_7) = {1}

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

            [cons](x1, x2) = [1] x1 + [1] x2 + [2]
                                                  
        [append^#](x1, x2) = [2] x1 + [3] x2 + [2]
                                                  
                 [c_5](x1) = [1] x1 + [1]         
                                                  
  [ifappend^#](x1, x2, x3) = [3] x2 + [2] x3 + [0]
                                                  
                 [c_7](x1) = [1] x1 + [0]         

This order satisfies following ordering constraints:

                [append^#(l1, l2)] = [2] l1 + [3] l2 + [2]        
                                   > [2] l1 + [3] l2 + [1]        
                                   = [c_5(ifappend^#(l1, l2, l1))]
                                                                  
  [ifappend^#(l1, l2, cons(x, l))] = [2] x + [2] l + [3] l2 + [4] 
                                   > [2] l + [3] l2 + [2]         
                                   = [c_7(append^#(l, l2))]       
                                                                  

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