YES(O(1),O(1))

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

Strict Trs:
  { not(x) -> xor(x, true())
  , implies(x, y) -> xor(and(x, y), xor(x, true()))
  , or(x, y) -> xor(and(x, y), xor(x, y))
  , =(x, y) -> xor(x, xor(y, true())) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(1))

We add the following weak dependency pairs:

Strict DPs:
  { not^#(x) -> c_1(x)
  , implies^#(x, y) -> c_2(x, y, x)
  , or^#(x, y) -> c_3(x, y, x, y)
  , =^#(x, y) -> c_4(x, y) }

and mark the set of starting terms.

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

Strict DPs:
  { not^#(x) -> c_1(x)
  , implies^#(x, y) -> c_2(x, y, x)
  , or^#(x, y) -> c_3(x, y, x, y)
  , =^#(x, y) -> c_4(x, y) }
Strict Trs:
  { not(x) -> xor(x, true())
  , implies(x, y) -> xor(and(x, y), xor(x, true()))
  , or(x, y) -> xor(and(x, y), xor(x, y))
  , =(x, y) -> xor(x, xor(y, true())) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(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(1)).

Strict DPs:
  { not^#(x) -> c_1(x)
  , implies^#(x, y) -> c_2(x, y, x)
  , or^#(x, y) -> c_3(x, y, x, y)
  , =^#(x, y) -> c_4(x, y) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(1))

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

The following argument positions are usable:
  none

TcT has computed the following constructor-restricted matrix
interpretation.

            [not^#](x1) = [1 1] x1 + [1]                               
                          [1 1]      [1]                               
                                                                       
              [c_1](x1) = [1 1] x1 + [0]                               
                          [1 1]      [1]                               
                                                                       
    [implies^#](x1, x2) = [2 2] x1 + [1 1] x2 + [1]                    
                          [2 1]      [1 1]      [1]                    
                                                                       
      [c_2](x1, x2, x3) = [1 1] x1 + [1 1] x2 + [1 1] x3 + [0]         
                          [1 2]      [1 1]      [2 1]      [1]         
                                                                       
         [or^#](x1, x2) = [2 2] x1 + [1 2] x2 + [1]                    
                          [1 2]      [2 1]      [1]                    
                                                                       
  [c_3](x1, x2, x3, x4) = [1 1] x1 + [0 1] x2 + [1 1] x3 + [1          
                                                            1] x4 + [0]
                          [2 2]      [2 2]      [1 1]      [2          
                                                            2]      [1]
                                                                       
          [=^#](x1, x2) = [1 1] x1 + [1 1] x2 + [2]                    
                          [1 1]      [1 1]      [1]                    
                                                                       
          [c_4](x1, x2) = [1 1] x1 + [1 1] x2 + [1]                    
                          [1 1]      [1 1]      [1]                    

The following symbols are considered usable

  {not^#, implies^#, or^#, =^#}

The order satisfies the following ordering constraints:

         [not^#(x)] = [1 1] x + [1]          
                      [1 1]     [1]          
                    > [1 1] x + [0]          
                      [1 1]     [1]          
                    = [c_1(x)]               
                                             
  [implies^#(x, y)] = [2 2] x + [1 1] y + [1]
                      [2 1]     [1 1]     [1]
                    ? [2 2] x + [1 1] y + [0]
                      [3 3]     [1 1]     [1]
                    = [c_2(x, y, x)]         
                                             
       [or^#(x, y)] = [2 2] x + [1 2] y + [1]
                      [1 2]     [2 1]     [1]
                    ? [2 2] x + [1 2] y + [0]
                      [3 3]     [4 4]     [1]
                    = [c_3(x, y, x, y)]      
                                             
        [=^#(x, y)] = [1 1] x + [1 1] y + [2]
                      [1 1]     [1 1]     [1]
                    > [1 1] x + [1 1] y + [1]
                      [1 1]     [1 1]     [1]
                    = [c_4(x, 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(1)).

Strict DPs:
  { implies^#(x, y) -> c_2(x, y, x)
  , or^#(x, y) -> c_3(x, y, x, y) }
Weak DPs:
  { not^#(x) -> c_1(x)
  , =^#(x, y) -> c_4(x, y) }
Obligation:
  runtime complexity
Answer:
  YES(O(1),O(1))

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

DPs:
  { 1: implies^#(x, y) -> c_2(x, y, x)
  , 2: or^#(x, y) -> c_3(x, y, x, y)
  , 3: not^#(x) -> c_1(x)
  , 4: =^#(x, y) -> c_4(x, y) }

Sub-proof:
----------
  The following argument positions are usable:
    none
  
  TcT has computed the following constructor-restricted matrix
  interpretation. Note that the diagonal of the component-wise maxima
  of interpretation-entries (of constructors) contains no more than 0
  non-zero entries.
  
                [not](x1) = [0]                                    
                                                                   
            [xor](x1, x2) = [0]                                    
                                                                   
                   [true] = [0]                                    
                                                                   
        [implies](x1, x2) = [0]                                    
                                                                   
            [and](x1, x2) = [0]                                    
                                                                   
             [or](x1, x2) = [0]                                    
                                                                   
              [=](x1, x2) = [0]                                    
                                                                   
              [not^#](x1) = [7] x1 + [7]                           
                                                                   
                [c_1](x1) = [7] x1 + [3]                           
                                                                   
      [implies^#](x1, x2) = [7] x1 + [7] x2 + [7]                  
                                                                   
        [c_2](x1, x2, x3) = [3] x1 + [7] x2 + [3] x3 + [3]         
                                                                   
           [or^#](x1, x2) = [7] x1 + [7] x2 + [7]                  
                                                                   
    [c_3](x1, x2, x3, x4) = [3] x1 + [3] x2 + [3] x3 + [3] x4 + [3]
                                                                   
            [=^#](x1, x2) = [7] x1 + [7] x2 + [7]                  
                                                                   
            [c_4](x1, x2) = [7] x1 + [7] x2 + [3]                  
  
  The following symbols are considered usable
  
    {not^#, implies^#, or^#, =^#}
  
  The order satisfies the following ordering constraints:
  
           [not^#(x)] = [7] x + [7]        
                      > [7] x + [3]        
                      = [c_1(x)]           
                                           
    [implies^#(x, y)] = [7] x + [7] y + [7]
                      > [6] x + [7] y + [3]
                      = [c_2(x, y, x)]     
                                           
         [or^#(x, y)] = [7] x + [7] y + [7]
                      > [6] x + [6] y + [3]
                      = [c_3(x, y, x, y)]  
                                           
          [=^#(x, y)] = [7] x + [7] y + [7]
                      > [7] x + [7] y + [3]
                      = [c_4(x, 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:
  { not^#(x) -> c_1(x)
  , implies^#(x, y) -> c_2(x, y, x)
  , or^#(x, y) -> c_3(x, y, x, y)
  , =^#(x, y) -> c_4(x, 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.

{ not^#(x) -> c_1(x)
, implies^#(x, y) -> c_2(x, y, x)
, or^#(x, y) -> c_3(x, y, x, y)
, =^#(x, y) -> c_4(x, 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(1))