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 the following innermost 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 the following constructor-restricted matrix
interpretation.

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

The following symbols are considered usable

  {D^#}

The order satisfies the following ordering constraints:

         [D^#(t())] = [1]                  
                      [0]                  
                    > [0]                  
                      [0]                  
                    = [c_1()]              
                                           
  [D^#(constant())] = [1]                  
                      [0]                  
                    > [0]                  
                      [0]                  
                    = [c_2()]              
                                           
     [D^#(+(x, y))] = [1]                  
                      [0]                  
                    ? [4]                  
                      [0]                  
                    = [c_3(D^#(x), D^#(y))]
                                           
     [D^#(*(x, y))] = [1]                  
                      [0]                  
                    ? [4]                  
                      [0]                  
                    = [c_4(D^#(x), D^#(y))]
                                           
     [D^#(-(x, y))] = [1]                  
                      [0]                  
                    ? [4]                  
                      [2]                  
                    = [c_5(D^#(x), D^#(y))]
                                           

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:
  { 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(1),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(1),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(1),O(n^1))

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

DPs:
  { 1: D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
  , 2: D^#(*(x, y)) -> c_4(D^#(x), D^#(y))
  , 3: D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }

Sub-proof:
----------
  The following argument positions are usable:
    Uargs(c_3) = {1, 2}, Uargs(c_4) = {1, 2}, Uargs(c_5) = {1, 2}
  
  TcT has computed the following constructor-based matrix
  interpretation satisfying not(EDA).
  
          [D](x1) = [7] x1 + [0]         
                                         
              [t] = [0]                  
                                         
              [1] = [0]                  
                                         
       [constant] = [0]                  
                                         
              [0] = [0]                  
                                         
      [+](x1, x2) = [1] x1 + [1] x2 + [4]
                                         
      [*](x1, x2) = [1] x1 + [1] x2 + [4]
                                         
      [-](x1, x2) = [1] x1 + [1] x2 + [4]
                                         
        [D^#](x1) = [2] x1 + [0]         
                                         
            [c_1] = [0]                  
                                         
            [c_2] = [0]                  
                                         
    [c_3](x1, x2) = [1] x1 + [1] x2 + [1]
                                         
    [c_4](x1, x2) = [1] x1 + [1] x2 + [7]
                                         
    [c_5](x1, x2) = [1] x1 + [1] x2 + [3]
  
  The following symbols are considered usable
  
    {D^#}
  
  The order satisfies the following ordering constraints:
  
    [D^#(+(x, y))] = [2] x + [2] y + [8]  
                   > [2] x + [2] y + [1]  
                   = [c_3(D^#(x), D^#(y))]
                                          
    [D^#(*(x, y))] = [2] x + [2] y + [8]  
                   > [2] x + [2] y + [7]  
                   = [c_4(D^#(x), D^#(y))]
                                          
    [D^#(-(x, y))] = [2] x + [2] y + [8]  
                   > [2] x + [2] y + [3]  
                   = [c_5(D^#(x), D^#(y))]
                                          

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:
  { 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(1))

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

{ 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)) }

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

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

Empty rules are trivially bounded

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