*** 1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        =(x,y) -> xor(x,xor(y,true()))
        implies(x,y) -> xor(and(x,y),xor(x,true()))
        not(x) -> xor(x,true())
        or(x,y) -> xor(and(x,y),xor(x,y))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {=/2,implies/2,not/1,or/2} / {and/2,true/0,xor/2}
      Obligation:
        Full
        basic terms: {=,implies,not,or}/{and,true,xor}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following weak dependency pairs:
      
      Strict DPs
        =#(x,y) -> c_1(x,y)
        implies#(x,y) -> c_2(x,y,x)
        not#(x) -> c_3(x)
        or#(x,y) -> c_4(x,y,x,y)
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        =#(x,y) -> c_1(x,y)
        implies#(x,y) -> c_2(x,y,x)
        not#(x) -> c_3(x)
        or#(x,y) -> c_4(x,y,x,y)
      Strict TRS Rules:
        =(x,y) -> xor(x,xor(y,true()))
        implies(x,y) -> xor(and(x,y),xor(x,true()))
        not(x) -> xor(x,true())
        or(x,y) -> xor(and(x,y),xor(x,y))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
      Obligation:
        Full
        basic terms: {=#,implies#,not#,or#}/{and,true,xor}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        =#(x,y) -> c_1(x,y)
        implies#(x,y) -> c_2(x,y,x)
        not#(x) -> c_3(x)
        or#(x,y) -> c_4(x,y,x,y)
*** 1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        =#(x,y) -> c_1(x,y)
        implies#(x,y) -> c_2(x,y,x)
        not#(x) -> c_3(x)
        or#(x,y) -> c_4(x,y,x,y)
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
      Obligation:
        Full
        basic terms: {=#,implies#,not#,or#}/{and,true,xor}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 0, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following constant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          none
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
                 p(=) = [1] x2 + [0]
               p(and) = [0]         
           p(implies) = [0]         
               p(not) = [0]         
                p(or) = [1] x2 + [0]
              p(true) = [0]         
               p(xor) = [0]         
                p(=#) = [2] x1 + [2]
          p(implies#) = [0]         
              p(not#) = [2]         
               p(or#) = [1]         
               p(c_1) = [2] x1 + [0]
               p(c_2) = [0]         
               p(c_3) = [0]         
               p(c_4) = [0]         
        
        Following rules are strictly oriented:
         =#(x,y) = [2] x + [2] 
                 > [2] x + [0] 
                 = c_1(x,y)    
        
         not#(x) = [2]         
                 > [0]         
                 = c_3(x)      
        
        or#(x,y) = [1]         
                 > [0]         
                 = c_4(x,y,x,y)
        
        
        Following rules are (at-least) weakly oriented:
        implies#(x,y) =  [0]       
                      >= [0]       
                      =  c_2(x,y,x)
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        implies#(x,y) -> c_2(x,y,x)
      Strict TRS Rules:
        
      Weak DP Rules:
        =#(x,y) -> c_1(x,y)
        not#(x) -> c_3(x)
        or#(x,y) -> c_4(x,y,x,y)
      Weak TRS Rules:
        
      Signature:
        {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
      Obligation:
        Full
        basic terms: {=#,implies#,not#,or#}/{and,true,xor}
    Applied Processor:
      NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules, greedy = NoGreedy}
    Proof:
      We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
      The following argument positions are considered usable:
        none
      
      Following symbols are considered usable:
        {}
      TcT has computed the following interpretation:
               p(=) = [0]          
             p(and) = [0]          
         p(implies) = [0]          
             p(not) = [0]          
              p(or) = [1]          
            p(true) = [0]          
             p(xor) = [0]          
              p(=#) = [1] x2 + [0] 
        p(implies#) = [1] x1 + [10]
            p(not#) = [0]          
             p(or#) = [0]          
             p(c_1) = [1] x2 + [0] 
             p(c_2) = [0]          
             p(c_3) = [0]          
             p(c_4) = [0]          
      
      Following rules are strictly oriented:
      implies#(x,y) = [1] x + [10]
                    > [0]         
                    = c_2(x,y,x)  
      
      
      Following rules are (at-least) weakly oriented:
       =#(x,y) =  [1] y + [0] 
               >= [1] y + [0] 
               =  c_1(x,y)    
      
       not#(x) =  [0]         
               >= [0]         
               =  c_3(x)      
      
      or#(x,y) =  [0]         
               >= [0]         
               =  c_4(x,y,x,y)
      
*** 1.1.1.1.1 Progress [(O(1),O(1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        =#(x,y) -> c_1(x,y)
        implies#(x,y) -> c_2(x,y,x)
        not#(x) -> c_3(x)
        or#(x,y) -> c_4(x,y,x,y)
      Weak TRS Rules:
        
      Signature:
        {=/2,implies/2,not/1,or/2,=#/2,implies#/2,not#/1,or#/2} / {and/2,true/0,xor/2,c_1/2,c_2/3,c_3/1,c_4/4}
      Obligation:
        Full
        basic terms: {=#,implies#,not#,or#}/{and,true,xor}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).