/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /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:
            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: 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:
            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: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:
            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: 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 1.b: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 1.b: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 1.b: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 1.b: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 1.b:6: NaturalMI.  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:
        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(c_3) = {1,2}
        
        Following symbols are considered usable:
          {mem#,or#}
        TcT has computed the following interpretation:
              p(=) = [1] x1 + [1] x2 + [1]
          p(false) = [1]                  
            p(mem) = [0]                  
            p(nil) = [4]                  
             p(or) = [2] x1 + [1] x2 + [1]
            p(set) = [1] x1 + [1]         
           p(true) = [2]                  
          p(union) = [1] x1 + [1] x2 + [8]
           p(mem#) = [1] x2 + [0]         
            p(or#) = [2] x2 + [1]         
            p(c_1) = [1]                  
            p(c_2) = [1]                  
            p(c_3) = [1] x1 + [1] x2 + [0]
            p(c_4) = [0]                  
            p(c_5) = [2]                  
            p(c_6) = [2]                  
        
        Following rules are strictly oriented:
        mem#(x,union(y,z)) = [1] y + [1] z + [8]     
                           > [1] y + [1] z + [0]     
                           = c_3(mem#(x,y),mem#(x,z))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b: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(Omega(n^1),O(n^1))