WORST_CASE(?,O(n^1))
* Step 1: WeightGap 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:
        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) = [0]                  
              p(nil) = [0]                  
               p(or) = [1] x1 + [1] x2 + [9]
              p(set) = [1] x1 + [0]         
             p(true) = [0]                  
            p(union) = [1] x1 + [1] x2 + [0]
          
          Following rules are strictly oriented:
                 or(x,true()) = [1] x + [9]
                              > [0]        
                              = true()     
          
          or(false(),false()) = [9]        
                              > [0]        
                              = false()    
          
                 or(true(),y) = [1] y + [9]
                              > [0]        
                              = 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]                  
                            >= [9]                  
                            =  or(mem(x,y),mem(x,z))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap 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))
        - Weak TRS:
            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:
        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) = [1]                  
              p(nil) = [8]                  
               p(or) = [1] x1 + [1] x2 + [1]
              p(set) = [8]                  
             p(true) = [8]                  
            p(union) = [1] x2 + [0]         
          
          Following rules are strictly oriented:
           mem(x,nil()) = [1]    
                        > [0]    
                        = false()
          
          mem(x,set(y)) = [1]    
                        > [0]    
                        = =(x,y) 
          
          
          Following rules are (at-least) weakly oriented:
            mem(x,union(y,z)) =  [1]                  
                              >= [3]                  
                              =  or(mem(x,y),mem(x,z))
          
                 or(x,true()) =  [1] x + [9]          
                              >= [8]                  
                              =  true()               
          
          or(false(),false()) =  [1]                  
                              >= [0]                  
                              =  false()              
          
                 or(true(),y) =  [1] y + [9]          
                              >= [8]                  
                              =  true()               
          
        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:
            mem(x,union(y,z)) -> or(mem(x,y),mem(x,z))
        - Weak TRS:
            mem(x,nil()) -> false()
            mem(x,set(y)) -> =(x,y)
            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:
        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(=) = [1]                  
            p(false) = [4]                  
              p(mem) = [2] x2 + [1]         
              p(nil) = [2]                  
               p(or) = [1] x1 + [1] x2 + [4]
              p(set) = [1] x1 + [2]         
             p(true) = [4]                  
            p(union) = [1] x1 + [1] x2 + [4]
          
          Following rules are strictly oriented:
          mem(x,union(y,z)) = [2] y + [2] z + [9]  
                            > [2] y + [2] z + [6]  
                            = or(mem(x,y),mem(x,z))
          
          
          Following rules are (at-least) weakly oriented:
                 mem(x,nil()) =  [5]        
                              >= [4]        
                              =  false()    
          
                mem(x,set(y)) =  [2] y + [5]
                              >= [1]        
                              =  =(x,y)     
          
                 or(x,true()) =  [1] x + [8]
                              >= [4]        
                              =  true()     
          
          or(false(),false()) =  [12]       
                              >= [4]        
                              =  false()    
          
                 or(true(),y) =  [1] y + [8]
                              >= [4]        
                              =  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))