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:
  { D(t()) -> 1()
  , D(constant()) -> 0()
  , D(+(x, y)) -> +(D(x), D(y))
  , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
  , D(-(x, y)) -> -(D(x), D(y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^1))

We add following weak dependency pairs:

Strict DPs:
  { D^#(t()) -> c_1()
  , D^#(constant()) -> c_2()
  , D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }

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:
  { D^#(t()) -> c_1()
  , D^#(constant()) -> c_2()
  , D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
Strict Trs:
  { D(t()) -> 1()
  , D(constant()) -> 0()
  , D(+(x, y)) -> +(D(x), D(y))
  , D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
  , D(-(x, y)) -> -(D(x), D(y)) }
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:
  { D^#(t()) -> c_1()
  , D^#(constant()) -> c_2()
  , D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
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_3) = {1, 2}, Uargs(c_4) = {1, 2}, Uargs(c_5) = {1, 2}

TcT has computed following constructor-restricted matrix
interpretation.

            [t] = [2]                  
                                       
     [constant] = [1]                  
                                       
    [+](x1, x2) = [1] x1 + [1] x2 + [1]
                                       
    [*](x1, x2) = [1] x1 + [1] x2 + [1]
                                       
    [-](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
      [D^#](x1) = [1]                  
                                       
          [c_1] = [0]                  
                                       
          [c_2] = [0]                  
                                       
  [c_3](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
  [c_4](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
  [c_5](x1, x2) = [1] x1 + [1] x2 + [2]

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:
  { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
Weak DPs:
  { D^#(t()) -> c_1()
  , D^#(constant()) -> c_2() }
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.

{ D^#(t()) -> c_1()
, D^#(constant()) -> c_2() }

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

Strict DPs:
  { D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
  , D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The following argument positions are usable:
  Uargs(c_3) = {1, 2}, Uargs(c_4) = {1, 2}, Uargs(c_5) = {1, 2}

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

    [+](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
    [*](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
    [-](x1, x2) = [1] x1 + [1] x2 + [2]
                                       
      [D^#](x1) = [2] x1 + [0]         
                                       
  [c_3](x1, x2) = [1] x1 + [1] x2 + [3]
                                       
  [c_4](x1, x2) = [1] x1 + [1] x2 + [3]
                                       
  [c_5](x1, x2) = [1] x1 + [1] x2 + [3]

This order satisfies following ordering constraints:

  [D^#(+(x, y))] = [2] x + [2] y + [4]  
                 > [2] x + [2] y + [3]  
                 = [c_3(D^#(x), D^#(y))]
                                        
  [D^#(*(x, y))] = [2] x + [2] y + [4]  
                 > [2] x + [2] y + [3]  
                 = [c_4(D^#(x), D^#(y))]
                                        
  [D^#(-(x, y))] = [2] x + [2] y + [4]  
                 > [2] x + [2] y + [3]  
                 = [c_5(D^#(x), D^#(y))]
                                        

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