* Step 1: MI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            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 runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):
        
        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] x_2 + [10]
              p(not) = [0]           
               p(or) = [1] x_2 + [10]
        
        Following rules are strictly oriented:
        implies(x,or(y,z)) = [2] z + [30]      
                           > [2] z + [20]      
                           = or(y,implies(x,z))
        
        
        Following rules are (at-least) weakly oriented:
              implies(not(x),y) =  [2] y + [10]      
                                >= [1] y + [10]      
                                =  or(x,y)           
        
        implies(not(x),or(y,z)) =  [2] z + [30]      
                                >= [2] z + [30]      
                                =  implies(y,or(x,z))
        
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            implies(not(x),y) -> or(x,y)
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Weak TRS:
            implies(x,or(y,z)) -> or(y,implies(x,z))
        - Signature:
            {implies/2} / {not/1,or/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        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:
            all
          TcT has computed the following interpretation:
            p(implies) = [2] x2 + [1]
                p(not) = [1] x1 + [0]
                 p(or) = [1] x2 + [0]
          
          Following rules are strictly oriented:
          implies(not(x),y) = [2] y + [1]
                            > [1] y + [0]
                            = or(x,y)    
          
          
          Following rules are (at-least) weakly oriented:
               implies(x,or(y,z)) =  [2] z + [1]       
                                  >= [2] z + [1]       
                                  =  or(y,implies(x,z))
          
          implies(not(x),or(y,z)) =  [2] z + [1]       
                                  >= [2] z + [1]       
                                  =  implies(y,or(x,z))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            implies(not(x),or(y,z)) -> implies(y,or(x,z))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        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:
            all
          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.
* Step 4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {implies} and constructors {not,or}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))