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)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x))
  , sqr(x) -> *(x, x) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the following weak dependency pairs:

Strict DPs:
  { sum^#(0()) -> c_1()
  , sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(x, x, sum^#(x))
  , sqr^#(x) -> c_4(x, x) }

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:
  { sum^#(0()) -> c_1()
  , sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(x, x, sum^#(x))
  , sqr^#(x) -> c_4(x, x) }
Strict Trs:
  { sum(0()) -> 0()
  , sum(s(x)) -> +(sqr(s(x)), sum(x))
  , sum(s(x)) -> +(*(s(x), s(x)), sum(x))
  , sqr(x) -> *(x, x) }
Obligation:
  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:
  { sum^#(0()) -> c_1()
  , sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(x, x, sum^#(x))
  , sqr^#(x) -> c_4(x, x) }
Obligation:
  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_2) = {1, 2}, Uargs(c_3) = {3}

TcT has computed the following constructor-restricted matrix
interpretation.

                [0] = [0]                                 
                      [0]                                 
                                                          
            [s](x1) = [0]                                 
                      [0]                                 
                                                          
        [sum^#](x1) = [0]                                 
                      [0]                                 
                                                          
              [c_1] = [1]                                 
                      [0]                                 
                                                          
      [c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [0]           
                      [0 1]      [0 1]      [0]           
                                                          
        [sqr^#](x1) = [1]                                 
                      [1]                                 
                                                          
  [c_3](x1, x2, x3) = [0 0] x1 + [0 0] x2 + [1 0] x3 + [2]
                      [1 2]      [2 2]      [0 1]      [0]
                                                          
      [c_4](x1, x2) = [0]                                 
                      [1]                                 

The following symbols are considered usable

  {sum^#, sqr^#}

The order satisfies the following ordering constraints:

   [sum^#(0())] = [0]                         
                  [0]                         
                ? [1]                         
                  [0]                         
                = [c_1()]                     
                                              
  [sum^#(s(x))] = [0]                         
                  [0]                         
                ? [1]                         
                  [1]                         
                = [c_2(sqr^#(s(x)), sum^#(x))]
                                              
  [sum^#(s(x))] = [0]                         
                  [0]                         
                ? [0 0] x + [2]               
                  [3 4]     [0]               
                = [c_3(x, x, sum^#(x))]       
                                              
     [sqr^#(x)] = [1]                         
                  [1]                         
                > [0]                         
                  [1]                         
                = [c_4(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 DPs:
  { sum^#(0()) -> c_1()
  , sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(x, x, sum^#(x)) }
Weak DPs: { sqr^#(x) -> c_4(x, x) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

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

DPs:
  { 1: sum^#(0()) -> c_1()
  , 2: sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , 3: sum^#(s(x)) -> c_3(x, x, sum^#(x)) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_2) = {1, 2}, Uargs(c_3) = {3}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
            [sum](x1) = [7] x1 + [0]         
                                             
                  [0] = [1]                  
                                             
              [s](x1) = [1] x1 + [4]         
                                             
          [+](x1, x2) = [1] x1 + [1] x2 + [0]
                                             
            [sqr](x1) = [7] x1 + [0]         
                                             
          [*](x1, x2) = [1] x1 + [1] x2 + [0]
                                             
          [sum^#](x1) = [2] x1 + [0]         
                                             
                [c_1] = [0]                  
                                             
        [c_2](x1, x2) = [4] x1 + [1] x2 + [7]
                                             
          [sqr^#](x1) = [0]                  
                                             
    [c_3](x1, x2, x3) = [1] x3 + [7]         
                                             
        [c_4](x1, x2) = [0]                  
  
  The following symbols are considered usable
  
    {sum^#, sqr^#}
  
  The order satisfies the following ordering constraints:
  
     [sum^#(0())] =  [2]                         
                  >  [0]                         
                  =  [c_1()]                     
                                                 
    [sum^#(s(x))] =  [2] x + [8]                 
                  >  [2] x + [7]                 
                  =  [c_2(sqr^#(s(x)), sum^#(x))]
                                                 
    [sum^#(s(x))] =  [2] x + [8]                 
                  >  [2] x + [7]                 
                  =  [c_3(x, x, sum^#(x))]       
                                                 
       [sqr^#(x)] =  [0]                         
                  >= [0]                         
                  =  [c_4(x, x)]                 
                                                 

The strictly oriented rules are moved into the weak component.

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

Weak DPs:
  { sum^#(0()) -> c_1()
  , sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
  , sum^#(s(x)) -> c_3(x, x, sum^#(x))
  , sqr^#(x) -> c_4(x, x) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(1))

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

{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, sum^#(s(x)) -> c_3(x, x, sum^#(x))
, sqr^#(x) -> c_4(x, x) }

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

Rules: Empty
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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