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:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sum(x), s(x))
  , sum1(0()) -> 0()
  , sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
Obligation:
  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}, Uargs(+) = {1}

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

    [sum](x1) = [1] x1 + [0]
                            
          [0] = [0]         
                            
      [s](x1) = [1] x1 + [1]
                            
  [+](x1, x2) = [1] x1 + [0]
                            
   [sum1](x1) = [0]         

The following symbols are considered usable

  {sum, sum1}

The order satisfies the following ordering constraints:

    [sum(0())] =  [0]                     
               >= [0]                     
               =  [0()]                   
                                          
   [sum(s(x))] =  [1] x + [1]             
               >  [1] x + [0]             
               =  [+(sum(x), s(x))]       
                                          
   [sum1(0())] =  [0]                     
               >= [0]                     
               =  [0()]                   
                                          
  [sum1(s(x))] =  [0]                     
               ?  [1]                     
               =  [s(+(sum1(x), +(x, x)))]
                                          

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:
  { sum(0()) -> 0()
  , sum1(0()) -> 0()
  , sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
Weak Trs: { sum(s(x)) -> +(sum(x), s(x)) }
Obligation:
  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}, Uargs(+) = {1}

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

    [sum](x1) = [1] x1 + [1]
                            
          [0] = [0]         
                            
      [s](x1) = [1] x1 + [4]
                            
  [+](x1, x2) = [1] x1 + [0]
                            
   [sum1](x1) = [0]         

The following symbols are considered usable

  {sum, sum1}

The order satisfies the following ordering constraints:

    [sum(0())] =  [1]                     
               >  [0]                     
               =  [0()]                   
                                          
   [sum(s(x))] =  [1] x + [5]             
               >  [1] x + [1]             
               =  [+(sum(x), s(x))]       
                                          
   [sum1(0())] =  [0]                     
               >= [0]                     
               =  [0()]                   
                                          
  [sum1(s(x))] =  [0]                     
               ?  [4]                     
               =  [s(+(sum1(x), +(x, x)))]
                                          

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:
  { sum1(0()) -> 0()
  , sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
Weak Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sum(x), s(x)) }
Obligation:
  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}, Uargs(+) = {1}

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

    [sum](x1) = [1] x1 + [0]
                            
          [0] = [4]         
                            
      [s](x1) = [1] x1 + [4]
                            
  [+](x1, x2) = [1] x1 + [4]
                            
   [sum1](x1) = [1] x1 + [4]

The following symbols are considered usable

  {sum, sum1}

The order satisfies the following ordering constraints:

    [sum(0())] =  [4]                     
               >= [4]                     
               =  [0()]                   
                                          
   [sum(s(x))] =  [1] x + [4]             
               >= [1] x + [4]             
               =  [+(sum(x), s(x))]       
                                          
   [sum1(0())] =  [8]                     
               >  [4]                     
               =  [0()]                   
                                          
  [sum1(s(x))] =  [1] x + [8]             
               ?  [1] x + [12]            
               =  [s(+(sum1(x), +(x, x)))]
                                          

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: { sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
Weak Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sum(x), s(x))
  , sum1(0()) -> 0() }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

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

Trs: { sum1(s(x)) -> s(+(sum1(x), +(x, x))) }

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}, Uargs(+) = {1}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
      [sum](x1) = [0]         
                              
            [0] = [0]         
                              
        [s](x1) = [1] x1 + [4]
                              
    [+](x1, x2) = [1] x1 + [0]
                              
     [sum1](x1) = [2] x1 + [0]
  
  The following symbols are considered usable
  
    {sum, sum1}
  
  The order satisfies the following ordering constraints:
  
      [sum(0())] =  [0]                     
                 >= [0]                     
                 =  [0()]                   
                                            
     [sum(s(x))] =  [0]                     
                 >= [0]                     
                 =  [+(sum(x), s(x))]       
                                            
     [sum1(0())] =  [0]                     
                 >= [0]                     
                 =  [0()]                   
                                            
    [sum1(s(x))] =  [2] x + [8]             
                 >  [2] x + [4]             
                 =  [s(+(sum1(x), +(x, x)))]
                                            

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:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sum(x), s(x))
  , sum1(0()) -> 0()
  , sum1(s(x)) -> s(+(sum1(x), +(x, x))) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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