/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1),O(n^1))
* 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:
             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: Sum.  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:
             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:2: 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:
             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: 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:
             runtime complexity wrt. defined symbols {mem,or} and constructors {=,false,nil,set,true,union}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          mem#(x,nil()) -> c_1()
          mem#(x,set(y)) -> c_2(x,y)
          mem#(x,union(y,z)) -> c_3(or#(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 1.b:2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            mem#(x,nil()) -> c_1()
            mem#(x,set(y)) -> c_2(x,y)
            mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)))
            or#(x,true()) -> c_4()
            or#(false(),false()) -> c_5()
            or#(true(),y) -> c_6()
        - 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,mem#/2,or#/2} / {=/2,false/0,nil/0,set/1,true/0,union/2,c_1/0,c_2/2,c_3/1,c_4/0,c_5/0,c_6/0}
        - Obligation:
             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 = WgOnTrs}
    + 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(or) = {1,2},
            uargs(or#) = {1,2},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                p(=) = [2]                   
            p(false) = [8]                   
              p(mem) = [1] x2 + [0]          
              p(nil) = [9]                   
               p(or) = [1] x1 + [1] x2 + [10]
              p(set) = [11]                  
             p(true) = [11]                  
            p(union) = [1] x1 + [1] x2 + [11]
             p(mem#) = [1] x2 + [0]          
              p(or#) = [1] x1 + [1] x2 + [0] 
              p(c_1) = [0]                   
              p(c_2) = [0]                   
              p(c_3) = [1] x1 + [0]          
              p(c_4) = [0]                   
              p(c_5) = [0]                   
              p(c_6) = [0]                   
          
          Following rules are strictly oriented:
                 mem#(x,nil()) = [9]                        
                               > [0]                        
                               = c_1()                      
          
                mem#(x,set(y)) = [11]                       
                               > [0]                        
                               = c_2(x,y)                   
          
            mem#(x,union(y,z)) = [1] y + [1] z + [11]       
                               > [1] y + [1] z + [0]        
                               = c_3(or#(mem(x,y),mem(x,z)))
          
                 or#(x,true()) = [1] x + [11]               
                               > [0]                        
                               = c_4()                      
          
          or#(false(),false()) = [16]                       
                               > [0]                        
                               = c_5()                      
          
                 or#(true(),y) = [1] y + [11]               
                               > [0]                        
                               = c_6()                      
          
                  mem(x,nil()) = [9]                        
                               > [8]                        
                               = false()                    
          
                 mem(x,set(y)) = [11]                       
                               > [2]                        
                               = =(x,y)                     
          
             mem(x,union(y,z)) = [1] y + [1] z + [11]       
                               > [1] y + [1] z + [10]       
                               = or(mem(x,y),mem(x,z))      
          
                  or(x,true()) = [1] x + [21]               
                               > [11]                       
                               = true()                     
          
           or(false(),false()) = [26]                       
                               > [8]                        
                               = false()                    
          
                  or(true(),y) = [1] y + [21]               
                               > [11]                       
                               = 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:3: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            mem#(x,nil()) -> c_1()
            mem#(x,set(y)) -> c_2(x,y)
            mem#(x,union(y,z)) -> c_3(or#(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_2/2,c_3/1,c_4/0,c_5/0,c_6/0}
        - Obligation:
             runtime complexity wrt. defined symbols {mem#,or#} and constructors {=,false,nil,set,true,union}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:mem#(x,nil()) -> c_1()
             
          
          2:W:mem#(x,set(y)) -> c_2(x,y)
             -->_2 mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z))):3
             -->_1 mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z))):3
             -->_2 or#(true(),y) -> c_6():6
             -->_1 or#(true(),y) -> c_6():6
             -->_2 or#(false(),false()) -> c_5():5
             -->_1 or#(false(),false()) -> c_5():5
             -->_2 or#(x,true()) -> c_4():4
             -->_1 or#(x,true()) -> c_4():4
             -->_2 mem#(x,set(y)) -> c_2(x,y):2
             -->_1 mem#(x,set(y)) -> c_2(x,y):2
             -->_2 mem#(x,nil()) -> c_1():1
             -->_1 mem#(x,nil()) -> c_1():1
          
          3:W:mem#(x,union(y,z)) -> c_3(or#(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
          
          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,set(y)) -> c_2(x,y)
          3: mem#(x,union(y,z)) -> c_3(or#(mem(x,y),mem(x,z)))
          4: or#(x,true()) -> c_4()
          5: or#(false(),false()) -> c_5()
          6: or#(true(),y) -> c_6()
          1: mem#(x,nil()) -> c_1()
** Step 1.b: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,mem#/2,or#/2} / {=/2,false/0,nil/0,set/1,true/0,union/2,c_1/0,c_2/2,c_3/1,c_4/0,c_5/0,c_6/0}
        - Obligation:
             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))