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:
  runtime complexity
Answer:
  YES(O(1),O(n^1))

We add the 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(y, D^#(x), 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(y, D^#(x), 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:
  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(y, D^#(x), x, D^#(y))
  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
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_3) = {1, 2}, Uargs(c_4) = {2, 4}, 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, x3, x4) = [1 0] x2 + [1 0] x4 + [2]
                          [0 1]      [0 1]      [0]
                                                   
          [c_5](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                          [0 1]      [0 1]      [0]

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(y, D^#(x), x, D^#(y))]
                                                 
     [D^#(-(x, y))] = [1]                        
                      [0]                        
                    ? [4]                        
                      [0]                        
                    = [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(y, D^#(x), x, D^#(y))
  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
Weak DPs:
  { D^#(t()) -> c_1()
  , D^#(constant()) -> c_2() }
Obligation:
  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(y, D^#(x), x, D^#(y))
  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
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: D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
  , 2: D^#(*(x, y)) -> c_4(y, D^#(x), 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) = {2, 4}, Uargs(c_5) = {1, 2}
  
  TcT has computed the 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) = [4] x1 + [0]         
                                                 
            [c_3](x1, x2) = [1] x1 + [1] x2 + [1]
                                                 
    [c_4](x1, x2, x3, x4) = [1] x2 + [1] x4 + [0]
                                                 
            [c_5](x1, x2) = [1] x1 + [1] x2 + [0]
  
  The order satisfies the following ordering constraints:
  
    [D^#(+(x, y))] = [4] x + [4] y + [8]        
                   > [4] x + [4] y + [1]        
                   = [c_3(D^#(x), D^#(y))]      
                                                
    [D^#(*(x, y))] = [4] x + [4] y + [8]        
                   > [4] x + [4] y + [0]        
                   = [c_4(y, D^#(x), x, D^#(y))]
                                                
    [D^#(-(x, y))] = [4] x + [4] y + [8]        
                   > [4] x + [4] y + [0]        
                   = [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(y, D^#(x), x, D^#(y))
  , D^#(-(x, y)) -> c_5(D^#(x), D^#(y)) }
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.

{ D^#(+(x, y)) -> c_3(D^#(x), D^#(y))
, D^#(*(x, y)) -> c_4(y, D^#(x), 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:
  runtime complexity
Answer:
  YES(O(1),O(1))

Empty rules are trivially bounded

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