WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs 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:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          mem#(x,nil()) -> c_1()
          mem#(x,set(y)) -> c_2()
          mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z))
          or#(x,true()) -> c_4()
          or#(false(),false()) -> c_5()
          or#(true(),y) -> c_6()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mem#(x,nil()) -> c_1()
            mem#(x,set(y)) -> c_2()
            mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z))
            or#(x,true()) -> c_4()
            or#(false(),false()) -> c_5()
            or#(true(),y) -> c_6()
        - 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,mem#/2,or#/2} / {=/2,false/0,nil/0,set/1,true/0,union/2,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {mem#,or#} and constructors {=,false,nil,set,true,union}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,4,5,6}
        by application of
          Pre({1,2,4,5,6}) = {3}.
        Here rules are labelled as follows:
          1: mem#(x,nil()) -> c_1()
          2: mem#(x,set(y)) -> c_2()
          3: mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z))
          4: or#(x,true()) -> c_4()
          5: or#(false(),false()) -> c_5()
          6: or#(true(),y) -> c_6()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z))
        - Weak DPs:
            mem#(x,nil()) -> c_1()
            mem#(x,set(y)) -> c_2()
            or#(x,true()) -> c_4()
            or#(false(),false()) -> c_5()
            or#(true(),y) -> c_6()
        - 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,mem#/2,or#/2} / {=/2,false/0,nil/0,set/1,true/0,union/2,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {mem#,or#} and constructors {=,false,nil,set,true,union}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z))
             -->_1 or#(true(),y) -> c_6():6
             -->_1 or#(false(),false()) -> c_5():5
             -->_1 or#(x,true()) -> c_4():4
             -->_3 mem#(x,set(y)) -> c_2():3
             -->_2 mem#(x,set(y)) -> c_2():3
             -->_3 mem#(x,nil()) -> c_1():2
             -->_2 mem#(x,nil()) -> c_1():2
             -->_3 mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z)):1
             -->_2 mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z)):1
          
          2:W:mem#(x,nil()) -> c_1()
             
          
          3:W:mem#(x,set(y)) -> c_2()
             
          
          4:W:or#(x,true()) -> c_4()
             
          
          5:W:or#(false(),false()) -> c_5()
             
          
          6:W:or#(true(),y) -> c_6()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: mem#(x,nil()) -> c_1()
          3: mem#(x,set(y)) -> c_2()
          4: or#(x,true()) -> c_4()
          5: or#(false(),false()) -> c_5()
          6: or#(true(),y) -> c_6()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z))
        - 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,mem#/2,or#/2} / {=/2,false/0,nil/0,set/1,true/0,union/2,c_1/0,c_2/0,c_3/3,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {mem#,or#} and constructors {=,false,nil,set,true,union}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z))
             -->_3 mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z)):1
             -->_2 mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)),mem#(x,y),mem#(x,z)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          mem#(x,union(y,z)) -> c_3(mem#(x,y),mem#(x,z))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mem#(x,union(y,z)) -> c_3(mem#(x,y),mem#(x,z))
        - 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,mem#/2,or#/2} / {=/2,false/0,nil/0,set/1,true/0,union/2,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {mem#,or#} and constructors {=,false,nil,set,true,union}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          mem#(x,union(y,z)) -> c_3(mem#(x,y),mem#(x,z))
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mem#(x,union(y,z)) -> c_3(mem#(x,y),mem#(x,z))
        - Signature:
            {mem/2,or/2,mem#/2,or#/2} / {=/2,false/0,nil/0,set/1,true/0,union/2,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0}
        - 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 constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(c_3) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(=) = [1] x1 + [1] x2 + [0]
            p(false) = [0]                  
              p(mem) = [0]                  
              p(nil) = [0]                  
               p(or) = [0]                  
              p(set) = [1] x1 + [0]         
             p(true) = [0]                  
            p(union) = [1] x1 + [1] x2 + [1]
             p(mem#) = [3] x2 + [0]         
              p(or#) = [0]                  
              p(c_1) = [0]                  
              p(c_2) = [0]                  
              p(c_3) = [1] x1 + [1] x2 + [0]
              p(c_4) = [0]                  
              p(c_5) = [0]                  
              p(c_6) = [0]                  
          
          Following rules are strictly oriented:
          mem#(x,union(y,z)) = [3] y + [3] z + [3]     
                             > [3] y + [3] z + [0]     
                             = c_3(mem#(x,y),mem#(x,z))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mem#(x,union(y,z)) -> c_3(mem#(x,y),mem#(x,z))
        - Signature:
            {mem/2,or/2,mem#/2,or#/2} / {=/2,false/0,nil/0,set/1,true/0,union/2,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0}
        - 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))