/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(?,O(n^1))
* Step 1: Sum.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  MAYBE
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(1)
          F (TrsFun "activate") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "cons") :: ["A"(0) x "A"(0)] -(0)-> "A"(0)
          F (TrsFun "from") :: ["A"(0)] -(1)-> "A"(0)
          F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "n__from") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1)
          F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "sel") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          activate(X) -> X
          activate(n__from(X)) -> from(X)
          from(X) -> cons(X,n__from(s(X)))
          from(X) -> n__from(X)
          sel(0(),cons(X,Y)) -> X
          sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
          main(x1,x2) -> sel(x1,x2)
        2. Weak:
          
        

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2} / {0/0,cons/2,n__from/1,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate,from,sel} and constructors {0,cons,n__from,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          activate#(X) -> c_1()
          activate#(n__from(X)) -> c_2(from#(X))
          from#(X) -> c_3()
          from#(X) -> c_4()
          sel#(0(),cons(X,Y)) -> c_5()
          sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(X))
            from#(X) -> c_3()
            from#(X) -> c_4()
            sel#(0(),cons(X,Y)) -> c_5()
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4,5}
        by application of
          Pre({1,3,4,5}) = {2,6}.
        Here rules are labelled as follows:
          1: activate#(X) -> c_1()
          2: activate#(n__from(X)) -> c_2(from#(X))
          3: from#(X) -> c_3()
          4: from#(X) -> c_4()
          5: sel#(0(),cons(X,Y)) -> c_5()
          6: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z))
** Step 1.b:3: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(n__from(X)) -> c_2(from#(X))
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z))
        - Weak DPs:
            activate#(X) -> c_1()
            from#(X) -> c_3()
            from#(X) -> c_4()
            sel#(0(),cons(X,Y)) -> c_5()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {2}.
        Here rules are labelled as follows:
          1: activate#(n__from(X)) -> c_2(from#(X))
          2: sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z))
          3: activate#(X) -> c_1()
          4: from#(X) -> c_3()
          5: from#(X) -> c_4()
          6: sel#(0(),cons(X,Y)) -> c_5()
** Step 1.b:4: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z))
        - Weak DPs:
            activate#(X) -> c_1()
            activate#(n__from(X)) -> c_2(from#(X))
            from#(X) -> c_3()
            from#(X) -> c_4()
            sel#(0(),cons(X,Y)) -> c_5()
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z))
             -->_2 activate#(n__from(X)) -> c_2(from#(X)):3
             -->_1 sel#(0(),cons(X,Y)) -> c_5():6
             -->_2 activate#(X) -> c_1():2
             -->_1 sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z)):1
          
          2:W:activate#(X) -> c_1()
             
          
          3:W:activate#(n__from(X)) -> c_2(from#(X))
             -->_1 from#(X) -> c_4():5
             -->_1 from#(X) -> c_3():4
          
          4:W:from#(X) -> c_3()
             
          
          5:W:from#(X) -> c_4()
             
          
          6:W:sel#(0(),cons(X,Y)) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: activate#(X) -> c_1()
          6: sel#(0(),cons(X,Y)) -> c_5()
          3: activate#(n__from(X)) -> c_2(from#(X))
          4: from#(X) -> c_3()
          5: from#(X) -> c_4()
** Step 1.b:5: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z))
             -->_1 sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)),activate#(Z)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
** Step 1.b:6: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
            sel(0(),cons(X,Y)) -> X
            sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
        - Signature:
            {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          activate(n__from(X)) -> from(X)
          from(X) -> cons(X,n__from(s(X)))
          from(X) -> n__from(X)
          sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
** Step 1.b:7: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + 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_6) = {1}
        
        Following symbols are considered usable:
          {activate#,from#,sel#}
        TcT has computed the following interpretation:
                  p(0) = [8]         
           p(activate) = [4]         
               p(cons) = [1] x2 + [1]
               p(from) = [8]         
            p(n__from) = [0]         
                  p(s) = [1] x1 + [4]
                p(sel) = [1]         
          p(activate#) = [1]         
              p(from#) = [4]         
               p(sel#) = [4] x1 + [2]
                p(c_1) = [0]         
                p(c_2) = [1]         
                p(c_3) = [0]         
                p(c_4) = [0]         
                p(c_5) = [1]         
                p(c_6) = [1] x1 + [4]
        
        Following rules are strictly oriented:
        sel#(s(X),cons(Y,Z)) = [4] X + [18]            
                             > [4] X + [6]             
                             = c_6(sel#(X,activate(Z)))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:8: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            sel#(s(X),cons(Y,Z)) -> c_6(sel#(X,activate(Z)))
        - Weak TRS:
            activate(X) -> X
            activate(n__from(X)) -> from(X)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {activate/1,from/1,sel/2,activate#/1,from#/1,sel#/2} / {0/0,cons/2,n__from/1,s/1,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/0,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {activate#,from#,sel#} and constructors {0,cons,n__from,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))