We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict Trs:
  { *(x, +(y, z)) -> +(*(x, y), *(x, z))
  , *(x, 1()) -> x
  , *(+(x, y), z) -> +(*(x, z), *(y, z))
  , *(1(), y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We add the following weak dependency pairs:

Strict DPs:
  { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
  , *^#(x, 1()) -> c_2()
  , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z))
  , *^#(1(), y) -> c_4() }

and mark the set of starting terms.

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
  , *^#(x, 1()) -> c_2()
  , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z))
  , *^#(1(), y) -> c_4() }
Strict Trs:
  { *(x, +(y, z)) -> +(*(x, y), *(x, z))
  , *(x, 1()) -> x
  , *(+(x, y), z) -> +(*(x, z), *(y, z))
  , *(1(), y) -> y }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

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^2)).

Strict DPs:
  { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
  , *^#(x, 1()) -> c_2()
  , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z))
  , *^#(1(), y) -> c_4() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

The weightgap principle applies (using the following constant
growth matrix-interpretation)

The following argument positions are usable:
  Uargs(c_1) = {1, 2}, Uargs(c_3) = {1, 2}

TcT has computed the following constructor-restricted matrix
interpretation.

    [+](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
                  [0 0]      [0 0]      [0]
                                           
            [1] = [0]                      
                  [0]                      
                                           
  [*^#](x1, x2) = [2]                      
                  [0]                      
                                           
  [c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                  [0 1]      [0 1]      [2]
                                           
          [c_2] = [0]                      
                  [0]                      
                                           
  [c_3](x1, x2) = [1 0] x1 + [1 0] x2 + [2]
                  [0 1]      [0 1]      [2]
                                           
          [c_4] = [0]                      
                  [0]                      

The order satisfies the following ordering constraints:

  [*^#(x, +(y, z))] = [2]                        
                      [0]                        
                    ? [6]                        
                      [2]                        
                    = [c_1(*^#(x, y), *^#(x, z))]
                                                 
      [*^#(x, 1())] = [2]                        
                      [0]                        
                    > [0]                        
                      [0]                        
                    = [c_2()]                    
                                                 
  [*^#(+(x, y), z)] = [2]                        
                      [0]                        
                    ? [6]                        
                      [2]                        
                    = [c_3(*^#(x, z), *^#(y, z))]
                                                 
      [*^#(1(), y)] = [2]                        
                      [0]                        
                    > [0]                        
                      [0]                        
                    = [c_4()]                    
                                                 

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^2)).

Strict DPs:
  { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
  , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }
Weak DPs:
  { *^#(x, 1()) -> c_2()
  , *^#(1(), y) -> c_4() }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

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

{ *^#(x, 1()) -> c_2()
, *^#(1(), y) -> c_4() }

We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).

Strict DPs:
  { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
  , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(O(1),O(n^2))

We use the processor 'custom shape polynomial interpretation' to
orient following rules strictly.

DPs:
  { 1: *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
  , 2: *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }

Sub-proof:
----------
  The following argument positions are considered usable:
    Uargs(c_1) = {1, 2}, Uargs(c_3) = {1, 2}
  TcT has computed the following constructor-restricted polynomial
  interpretation.
    [+](x1, x2) = 2 + x1 + x2            
                                         
  [*^#](x1, x2) = 1 + 2*x1 + 2*x1*x2 + x2
                                         
  [c_1](x1, x2) = x1 + x2                
                                         
  [c_3](x1, x2) = 2 + x1 + x2            
                                         
  
  This order satisfies the following ordering constraints.
  
    [*^#(x, +(y, z))] = 3 + 6*x + 2*x*y + 2*x*z + y + z    
                      > 2 + 4*x + 2*x*y + y + 2*x*z + z    
                      = [c_1(*^#(x, y), *^#(x, z))]        
                                                           
    [*^#(+(x, y), z)] = 5 + 2*x + 2*y + 5*z + 2*x*z + 2*y*z
                      > 4 + 2*x + 2*x*z + 2*z + 2*y + 2*y*z
                      = [c_3(*^#(x, z), *^#(y, z))]        
                                                           

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:
  { *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
  , *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }
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.

{ *^#(x, +(y, z)) -> c_1(*^#(x, y), *^#(x, z))
, *^#(+(x, y), z) -> c_3(*^#(x, z), *^#(y, z)) }

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^2))