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:
  { average(x, s(s(s(y)))) -> s(average(s(x), y))
  , average(s(x), y) -> average(x, s(y))
  , average(0(), s(s(0()))) -> s(0())
  , average(0(), s(0())) -> 0()
  , average(0(), 0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(s) = {1}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

  [average](x1, x2) = [1]         
                                  
            [s](x1) = [1] x1 + [0]
                                  
                [0] = [0]         

The following symbols are considered usable

  {average}

The order satisfies the following ordering constraints:

   [average(x, s(s(s(y))))] =  [1]                  
                            >= [1]                  
                            =  [s(average(s(x), y))]
                                                    
         [average(s(x), y)] =  [1]                  
                            >= [1]                  
                            =  [average(x, s(y))]   
                                                    
  [average(0(), s(s(0())))] =  [1]                  
                            >  [0]                  
                            =  [s(0())]             
                                                    
     [average(0(), s(0()))] =  [1]                  
                            >  [0]                  
                            =  [0()]                
                                                    
        [average(0(), 0())] =  [1]                  
                            >  [0]                  
                            =  [0()]                
                                                    

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(n^1)).

Strict Trs:
  { average(x, s(s(s(y)))) -> s(average(s(x), y))
  , average(s(x), y) -> average(x, s(y)) }
Weak Trs:
  { average(0(), s(s(0()))) -> s(0())
  , average(0(), s(0())) -> 0()
  , average(0(), 0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(s) = {1}

TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).

  [average](x1, x2) = [1] x1 + [1] x2 + [0]
                                           
            [s](x1) = [1] x1 + [4]         
                                           
                [0] = [0]                  

The following symbols are considered usable

  {average}

The order satisfies the following ordering constraints:

   [average(x, s(s(s(y))))] =  [1] x + [1] y + [12] 
                            >  [1] x + [1] y + [8]  
                            =  [s(average(s(x), y))]
                                                    
         [average(s(x), y)] =  [1] x + [1] y + [4]  
                            >= [1] x + [1] y + [4]  
                            =  [average(x, s(y))]   
                                                    
  [average(0(), s(s(0())))] =  [8]                  
                            >  [4]                  
                            =  [s(0())]             
                                                    
     [average(0(), s(0()))] =  [4]                  
                            >  [0]                  
                            =  [0()]                
                                                    
        [average(0(), 0())] =  [0]                  
                            >= [0]                  
                            =  [0()]                
                                                    

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(n^1)).

Strict Trs: { average(s(x), y) -> average(x, s(y)) }
Weak Trs:
  { average(x, s(s(s(y)))) -> s(average(s(x), y))
  , average(0(), s(s(0()))) -> s(0())
  , average(0(), s(0())) -> 0()
  , average(0(), 0()) -> 0() }
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.

Trs: { average(s(x), y) -> average(x, s(y)) }

The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(s) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
    [average](x1, x2) = [2] x1 + [1] x2 + [6]
                                             
              [s](x1) = [1] x1 + [1]         
                                             
                  [0] = [0]                  
  
  The following symbols are considered usable
  
    {average}
  
  The order satisfies the following ordering constraints:
  
     [average(x, s(s(s(y))))] =  [2] x + [1] y + [9]  
                              >= [2] x + [1] y + [9]  
                              =  [s(average(s(x), y))]
                                                      
           [average(s(x), y)] =  [2] x + [1] y + [8]  
                              >  [2] x + [1] y + [7]  
                              =  [average(x, s(y))]   
                                                      
    [average(0(), s(s(0())))] =  [8]                  
                              >  [1]                  
                              =  [s(0())]             
                                                      
       [average(0(), s(0()))] =  [7]                  
                              >  [0]                  
                              =  [0()]                
                                                      
          [average(0(), 0())] =  [6]                  
                              >  [0]                  
                              =  [0()]                
                                                      

We return to the main proof.

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

Weak Trs:
  { average(x, s(s(s(y)))) -> s(average(s(x), y))
  , average(s(x), y) -> average(x, s(y))
  , average(0(), s(s(0()))) -> s(0())
  , average(0(), s(0())) -> 0()
  , average(0(), 0()) -> 0() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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