* Step 1: Sum WORST_CASE(Omega(n^1),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:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(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:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          mem(x,y){y -> union(y,z)} =
            mem(x,union(y,z)) ->^+ or(mem(x,y),mem(x,z))
              = C[mem(x,y) = mem(x,y){}]

** Step 1.b: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) = [3]                   
              p(mem) = [4] x2 + [4]          
              p(nil) = [1]                   
               p(or) = [1] x1 + [1] x2 + [12]
              p(set) = [3]                   
             p(true) = [8]                   
            p(union) = [1] x1 + [1] x2 + [6] 
          
          Following rules are strictly oriented:
                 mem(x,nil()) = [8]                  
                              > [3]                  
                              = false()              
          
                mem(x,set(y)) = [16]                 
                              > [0]                  
                              = =(x,y)               
          
            mem(x,union(y,z)) = [4] y + [4] z + [28] 
                              > [4] y + [4] z + [20] 
                              = or(mem(x,y),mem(x,z))
          
                 or(x,true()) = [1] x + [20]         
                              > [8]                  
                              = true()               
          
          or(false(),false()) = [18]                 
                              > [3]                  
                              = false()              
          
                 or(true(),y) = [1] y + [20]         
                              > [8]                  
                              = true()               
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: 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(Omega(n^1),O(n^1))