WORST_CASE(?,O(n^1))
* Step 1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            mem(x,nil()) -> false()
            mem(x,set(y)) -> =(x,y)
            mem(x,union(y,z)) -> or(mem(x,y),mem(x,z))
            or(x,true()) -> true()
            or(false(),false()) -> false()
            or(true(),y) -> true()
        - Signature:
            {mem/2,or/2} / {=/2,false/0,nil/0,set/1,true/0,union/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {mem,or} and constructors {=,false,nil,set,true,union}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(or) = {1,2}
        
        Following symbols are considered usable:
          {mem,or}
        TcT has computed the following interpretation:
              p(=) = [0]                  
          p(false) = [0]                  
            p(mem) = [0]                  
            p(nil) = [0]                  
             p(or) = [8] x1 + [8] x2 + [0]
            p(set) = [1] x1 + [0]         
           p(true) = [1]                  
          p(union) = [1] x2 + [2]         
        
        Following rules are strictly oriented:
        or(x,true()) = [8] x + [8]
                     > [1]        
                     = true()     
        
        or(true(),y) = [8] y + [8]
                     > [1]        
                     = true()     
        
        
        Following rules are (at-least) weakly oriented:
               mem(x,nil()) =  [0]                  
                            >= [0]                  
                            =  false()              
        
              mem(x,set(y)) =  [0]                  
                            >= [0]                  
                            =  =(x,y)               
        
          mem(x,union(y,z)) =  [0]                  
                            >= [0]                  
                            =  or(mem(x,y),mem(x,z))
        
        or(false(),false()) =  [0]                  
                            >= [0]                  
                            =  false()              
        
* Step 2: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            mem(x,nil()) -> false()
            mem(x,set(y)) -> =(x,y)
            mem(x,union(y,z)) -> or(mem(x,y),mem(x,z))
            or(false(),false()) -> false()
        - Weak TRS:
            or(x,true()) -> true()
            or(true(),y) -> true()
        - Signature:
            {mem/2,or/2} / {=/2,false/0,nil/0,set/1,true/0,union/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {mem,or} and constructors {=,false,nil,set,true,union}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(or) = {1,2}
        
        Following symbols are considered usable:
          {mem,or}
        TcT has computed the following interpretation:
              p(=) = [0]                  
          p(false) = [0]                  
            p(mem) = [2] x2 + [0]         
            p(nil) = [0]                  
             p(or) = [1] x1 + [1] x2 + [0]
            p(set) = [0]                  
           p(true) = [0]                  
          p(union) = [1] x1 + [1] x2 + [2]
        
        Following rules are strictly oriented:
        mem(x,union(y,z)) = [2] y + [2] z + [4]  
                          > [2] y + [2] z + [0]  
                          = or(mem(x,y),mem(x,z))
        
        
        Following rules are (at-least) weakly oriented:
               mem(x,nil()) =  [0]        
                            >= [0]        
                            =  false()    
        
              mem(x,set(y)) =  [0]        
                            >= [0]        
                            =  =(x,y)     
        
               or(x,true()) =  [1] x + [0]
                            >= [0]        
                            =  true()     
        
        or(false(),false()) =  [0]        
                            >= [0]        
                            =  false()    
        
               or(true(),y) =  [1] y + [0]
                            >= [0]        
                            =  true()     
        
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            mem(x,nil()) -> false()
            mem(x,set(y)) -> =(x,y)
            or(false(),false()) -> false()
        - Weak TRS:
            mem(x,union(y,z)) -> or(mem(x,y),mem(x,z))
            or(x,true()) -> true()
            or(true(),y) -> true()
        - Signature:
            {mem/2,or/2} / {=/2,false/0,nil/0,set/1,true/0,union/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {mem,or} and constructors {=,false,nil,set,true,union}
    + 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) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(=) = [0]                  
            p(false) = [0]                  
              p(mem) = [6] x2 + [4]         
              p(nil) = [0]                  
               p(or) = [1] x1 + [1] x2 + [2]
              p(set) = [2]                  
             p(true) = [14]                 
            p(union) = [1] x1 + [1] x2 + [1]
          
          Following rules are strictly oriented:
                 mem(x,nil()) = [4]    
                              > [0]    
                              = false()
          
                mem(x,set(y)) = [16]   
                              > [0]    
                              = =(x,y) 
          
          or(false(),false()) = [2]    
                              > [0]    
                              = false()
          
          
          Following rules are (at-least) weakly oriented:
          mem(x,union(y,z)) =  [6] y + [6] z + [10] 
                            >= [6] y + [6] z + [10] 
                            =  or(mem(x,y),mem(x,z))
          
               or(x,true()) =  [1] x + [16]         
                            >= [14]                 
                            =  true()               
          
               or(true(),y) =  [1] y + [16]         
                            >= [14]                 
                            =  true()               
          
        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:
            mem(x,nil()) -> false()
            mem(x,set(y)) -> =(x,y)
            mem(x,union(y,z)) -> or(mem(x,y),mem(x,z))
            or(x,true()) -> true()
            or(false(),false()) -> false()
            or(true(),y) -> true()
        - Signature:
            {mem/2,or/2} / {=/2,false/0,nil/0,set/1,true/0,union/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {mem,or} and constructors {=,false,nil,set,true,union}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))