*** 1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        implies(x,or(y,z)) -> or(y,implies(x,z))
        implies(not(x),y) -> or(x,y)
        implies(not(x),or(y,z)) -> implies(y,or(x,z))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {implies/2} / {not/1,or/2}
      Obligation:
        Innermost
        basic terms: {implies}/{not,or}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(or) = {2}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
          p(implies) = [2] x2 + [7]
              p(not) = [1] x1 + [0]
               p(or) = [1] x2 + [0]
        
        Following rules are strictly oriented:
        implies(not(x),y) = [2] y + [7]
                          > [1] y + [0]
                          = or(x,y)    
        
        
        Following rules are (at-least) weakly oriented:
             implies(x,or(y,z)) =  [2] z + [7]       
                                >= [2] z + [7]       
                                =  or(y,implies(x,z))
        
        implies(not(x),or(y,z)) =  [2] z + [7]       
                                >= [2] z + [7]       
                                =  implies(y,or(x,z))
        
      Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** 1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        implies(x,or(y,z)) -> or(y,implies(x,z))
        implies(not(x),or(y,z)) -> implies(y,or(x,z))
      Weak DP Rules:
        
      Weak TRS Rules:
        implies(not(x),y) -> or(x,y)
      Signature:
        {implies/2} / {not/1,or/2}
      Obligation:
        Innermost
        basic terms: {implies}/{not,or}
    Applied Processor:
      NaturalMI {miDimension = 1, miDegree = 1, 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:
      The following argument positions are considered usable:
        uargs(or) = {2}
      
      Following symbols are considered usable:
        {implies}
      TcT has computed the following interpretation:
        p(implies) = [2] x2 + [6]
            p(not) = [0]         
             p(or) = [1] x2 + [6]
      
      Following rules are strictly oriented:
      implies(x,or(y,z)) = [2] z + [18]      
                         > [2] z + [12]      
                         = or(y,implies(x,z))
      
      
      Following rules are (at-least) weakly oriented:
            implies(not(x),y) =  [2] y + [6]       
                              >= [1] y + [6]       
                              =  or(x,y)           
      
      implies(not(x),or(y,z)) =  [2] z + [18]      
                              >= [2] z + [18]      
                              =  implies(y,or(x,z))
      
*** 1.1.1 Progress [(O(1),O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        implies(not(x),or(y,z)) -> implies(y,or(x,z))
      Weak DP Rules:
        
      Weak TRS Rules:
        implies(x,or(y,z)) -> or(y,implies(x,z))
        implies(not(x),y) -> or(x,y)
      Signature:
        {implies/2} / {not/1,or/2}
      Obligation:
        Innermost
        basic terms: {implies}/{not,or}
    Applied Processor:
      WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    Proof:
      The weightgap principle applies using the following nonconstant growth matrix-interpretation:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(or) = {2}
        
        Following symbols are considered usable:
          {}
        TcT has computed the following interpretation:
          p(implies) = [1] x1 + [1] x2 + [0]
              p(not) = [1] x1 + [2]         
               p(or) = [1] x1 + [1] x2 + [1]
        
        Following rules are strictly oriented:
        implies(not(x),or(y,z)) = [1] x + [1] y + [1] z + [3]
                                > [1] x + [1] y + [1] z + [1]
                                = implies(y,or(x,z))         
        
        
        Following rules are (at-least) weakly oriented:
        implies(x,or(y,z)) =  [1] x + [1] y + [1] z + [1]
                           >= [1] x + [1] y + [1] z + [1]
                           =  or(y,implies(x,z))         
        
         implies(not(x),y) =  [1] x + [1] y + [2]        
                           >= [1] x + [1] y + [1]        
                           =  or(x,y)                    
        
      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:
        
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        implies(x,or(y,z)) -> or(y,implies(x,z))
        implies(not(x),y) -> or(x,y)
        implies(not(x),or(y,z)) -> implies(y,or(x,z))
      Signature:
        {implies/2} / {not/1,or/2}
      Obligation:
        Innermost
        basic terms: {implies}/{not,or}
    Applied Processor:
      EmptyProcessor
    Proof:
      The problem is already closed. The intended complexity is O(1).