YES(?,O(n^2))

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^2)).

Strict Trs:
  { f(0()) -> true()
  , f(1()) -> false()
  , f(s(x)) -> f(x)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , g(x, c(y)) -> g(x, g(s(c(y)), y))
  , g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

We add following dependency tuples:

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(1()) -> c_2()
  , f^#(s(x)) -> c_3(f^#(x))
  , if^#(true(), x, y) -> c_4()
  , if^#(false(), x, y) -> c_5()
  , g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y))
  , g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^2)).

Strict DPs:
  { f^#(0()) -> c_1()
  , f^#(1()) -> c_2()
  , f^#(s(x)) -> c_3(f^#(x))
  , if^#(true(), x, y) -> c_4()
  , if^#(false(), x, y) -> c_5()
  , g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y))
  , g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) }
Weak Trs:
  { f(0()) -> true()
  , f(1()) -> false()
  , f(s(x)) -> f(x)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , g(x, c(y)) -> g(x, g(s(c(y)), y))
  , g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

We estimate the number of application of {1,2,4,5} by applications
of Pre({1,2,4,5}) = {3,7}. Here rules are labeled as follows:

  DPs:
    { 1: f^#(0()) -> c_1()
    , 2: f^#(1()) -> c_2()
    , 3: f^#(s(x)) -> c_3(f^#(x))
    , 4: if^#(true(), x, y) -> c_4()
    , 5: if^#(false(), x, y) -> c_5()
    , 6: g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y))
    , 7: g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^2)).

Strict DPs:
  { f^#(s(x)) -> c_3(f^#(x))
  , g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y))
  , g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) }
Weak DPs:
  { f^#(0()) -> c_1()
  , f^#(1()) -> c_2()
  , if^#(true(), x, y) -> c_4()
  , if^#(false(), x, y) -> c_5() }
Weak Trs:
  { f(0()) -> true()
  , f(1()) -> false()
  , f(s(x)) -> f(x)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , g(x, c(y)) -> g(x, g(s(c(y)), y))
  , g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

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

{ f^#(0()) -> c_1()
, f^#(1()) -> c_2()
, if^#(true(), x, y) -> c_4()
, if^#(false(), x, y) -> c_5() }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^2)).

Strict DPs:
  { f^#(s(x)) -> c_3(f^#(x))
  , g^#(x, c(y)) -> c_6(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y))
  , g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) }
Weak Trs:
  { f(0()) -> true()
  , f(1()) -> false()
  , f(s(x)) -> f(x)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , g(x, c(y)) -> g(x, g(s(c(y)), y))
  , g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:

  { g^#(s(x), s(y)) -> c_7(if^#(f(x), s(x), s(y)), f^#(x)) }

We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^2)).

Strict DPs:
  { f^#(s(x)) -> c_1(f^#(x))
  , g^#(x, c(y)) -> c_2(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y))
  , g^#(s(x), s(y)) -> c_3(f^#(x)) }
Weak Trs:
  { f(0()) -> true()
  , f(1()) -> false()
  , f(s(x)) -> f(x)
  , if(true(), x, y) -> x
  , if(false(), x, y) -> y
  , g(x, c(y)) -> g(x, g(s(c(y)), y))
  , g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^2))

The following argument positions are usable:
  Uargs(c_1) = {1}, Uargs(c_2) = {1, 2}, Uargs(c_3) = {1}

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

           [f](x1) = [0]                      
                     [0]                      
                                              
               [0] = [0]                      
                     [0]                      
                                              
            [true] = [0]                      
                     [0]                      
                                              
               [1] = [0]                      
                     [0]                      
                                              
           [false] = [0]                      
                     [0]                      
                                              
           [s](x1) = [1 0] x1 + [2]           
                     [0 0]      [0]           
                                              
  [if](x1, x2, x3) = [2 0] x2 + [1 0] x3 + [0]
                     [0 2]      [0 2]      [0]
                                              
       [g](x1, x2) = [2 0] x1 + [1 2] x2 + [0]
                     [0 0]      [0 0]      [0]
                                              
           [c](x1) = [0 0] x1 + [0]           
                     [1 1]      [3]           
                                              
         [f^#](x1) = [1 0] x1 + [0]           
                     [0 0]      [1]           
                                              
     [g^#](x1, x2) = [1 0] x1 + [0 1] x2 + [0]
                     [0 0]      [0 0]      [0]
                                              
         [c_1](x1) = [1 0] x1 + [1]           
                     [0 0]      [1]           
                                              
     [c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                     [0 0]      [0 0]      [0]
                                              
         [c_3](x1) = [1 1] x1 + [0]           
                     [0 0]      [0]           

This order satisfies following ordering constraints:

   [if(true(), x, y)] =  [2 0] x + [1 0] y + [0]                      
                         [0 2]     [0 2]     [0]                      
                      >= [1 0] x + [0]                                
                         [0 1]     [0]                                
                      =  [x]                                          
                                                                      
  [if(false(), x, y)] =  [2 0] x + [1 0] y + [0]                      
                         [0 2]     [0 2]     [0]                      
                      >= [1 0] y + [0]                                
                         [0 1]     [0]                                
                      =  [y]                                          
                                                                      
         [g(x, c(y))] =  [2 0] x + [2 2] y + [6]                      
                         [0 0]     [0 0]     [0]                      
                      >  [2 0] x + [1 2] y + [4]                      
                         [0 0]     [0 0]     [0]                      
                      =  [g(x, g(s(c(y)), y))]                        
                                                                      
      [g(s(x), s(y))] =  [2 0] x + [1 0] y + [6]                      
                         [0 0]     [0 0]     [0]                      
                      >= [2 0] x + [1 0] y + [6]                      
                         [0 0]     [0 0]     [0]                      
                      =  [if(f(x), s(x), s(y))]                       
                                                                      
          [f^#(s(x))] =  [1 0] x + [2]                                
                         [0 0]     [1]                                
                      >  [1 0] x + [1]                                
                         [0 0]     [1]                                
                      =  [c_1(f^#(x))]                                
                                                                      
       [g^#(x, c(y))] =  [1 0] x + [1 1] y + [3]                      
                         [0 0]     [0 0]     [0]                      
                      >  [1 0] x + [0 1] y + [2]                      
                         [0 0]     [0 0]     [0]                      
                      =  [c_2(g^#(x, g(s(c(y)), y)), g^#(s(c(y)), y))]
                                                                      
    [g^#(s(x), s(y))] =  [1 0] x + [2]                                
                         [0 0]     [0]                                
                      >  [1 0] x + [1]                                
                         [0 0]     [0]                                
                      =  [c_3(f^#(x))]                                
                                                                      

Hurray, we answered YES(?,O(n^2))