/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(?,O(n^1))
* Step 1: Sum.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
            activate(X) -> X
            activate(n__cons(X1,X2)) -> cons(X1,X2)
            activate(n__from(X)) -> from(X)
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1} / {n__cons/2,n__from/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd,activate,cons,from} and constructors {n__cons,n__from,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
            activate(X) -> X
            activate(n__cons(X1,X2)) -> cons(X1,X2)
            activate(n__from(X)) -> from(X)
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1} / {n__cons/2,n__from/1,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd,activate,cons,from} and constructors {n__cons,n__from,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y))
          activate#(X) -> c_2(X)
          activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
          activate#(n__from(X)) -> c_4(from#(X))
          cons#(X1,X2) -> c_5(X1,X2)
          from#(X) -> c_6(cons#(X,n__from(s(X))))
          from#(X) -> c_7(X)
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            2nd#(cons(X,n__cons(Y,Z))) -> c_1(activate#(Y))
            activate#(X) -> c_2(X)
            activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            cons#(X1,X2) -> c_5(X1,X2)
            from#(X) -> c_6(cons#(X,n__from(s(X))))
            from#(X) -> c_7(X)
        - Strict TRS:
            2nd(cons(X,n__cons(Y,Z))) -> activate(Y)
            activate(X) -> X
            activate(n__cons(X1,X2)) -> cons(X1,X2)
            activate(n__from(X)) -> from(X)
            cons(X1,X2) -> n__cons(X1,X2)
            from(X) -> cons(X,n__from(s(X)))
            from(X) -> n__from(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#} and constructors {n__cons,n__from,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate#(X) -> c_2(X)
          activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
          activate#(n__from(X)) -> c_4(from#(X))
          cons#(X1,X2) -> c_5(X1,X2)
          from#(X) -> c_6(cons#(X,n__from(s(X))))
          from#(X) -> c_7(X)
* Step 4: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_2(X)
            activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            cons#(X1,X2) -> c_5(X1,X2)
            from#(X) -> c_6(cons#(X,n__from(s(X))))
            from#(X) -> c_7(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#} and constructors {n__cons,n__from,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: activate#(X) -> c_2(X)
          2: activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
          3: activate#(n__from(X)) -> c_4(from#(X))
          5: from#(X) -> c_6(cons#(X,n__from(s(X))))
          6: from#(X) -> c_7(X)
          
        The strictly oriented rules are moved into the weak component.
** Step 4.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            activate#(X) -> c_2(X)
            activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            cons#(X1,X2) -> c_5(X1,X2)
            from#(X) -> c_6(cons#(X,n__from(s(X))))
            from#(X) -> c_7(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#} and constructors {n__cons,n__from,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and 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},
          uargs(c_4) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                p(2nd) = [1] x1 + [2]         
           p(activate) = [2] x1 + [8]         
               p(cons) = [1] x1 + [1] x2 + [2]
               p(from) = [1]                  
            p(n__cons) = [1] x1 + [0]         
            p(n__from) = [1] x1 + [3]         
                  p(s) = [1] x1 + [10]        
               p(2nd#) = [1] x1 + [0]         
          p(activate#) = [12]                 
              p(cons#) = [0]                  
              p(from#) = [2]                  
                p(c_1) = [1]                  
                p(c_2) = [0]                  
                p(c_3) = [1] x1 + [1]         
                p(c_4) = [4] x1 + [1]         
                p(c_5) = [0]                  
                p(c_6) = [2] x1 + [0]         
                p(c_7) = [0]                  
        
        Following rules are strictly oriented:
                     activate#(X) = [12]                       
                                  > [0]                        
                                  = c_2(X)                     
        
        activate#(n__cons(X1,X2)) = [12]                       
                                  > [1]                        
                                  = c_3(cons#(X1,X2))          
        
            activate#(n__from(X)) = [12]                       
                                  > [9]                        
                                  = c_4(from#(X))              
        
                         from#(X) = [2]                        
                                  > [0]                        
                                  = c_6(cons#(X,n__from(s(X))))
        
                         from#(X) = [2]                        
                                  > [0]                        
                                  = c_7(X)                     
        
        
        Following rules are (at-least) weakly oriented:
        cons#(X1,X2) =  [0]       
                     >= [0]       
                     =  c_5(X1,X2)
        
** Step 4.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(X1,X2) -> c_5(X1,X2)
        - Weak DPs:
            activate#(X) -> c_2(X)
            activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            from#(X) -> c_6(cons#(X,n__from(s(X))))
            from#(X) -> c_7(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#} and constructors {n__cons,n__from,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

** Step 4.b:1: PredecessorEstimationCP.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(X1,X2) -> c_5(X1,X2)
        - Weak DPs:
            activate#(X) -> c_2(X)
            activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            from#(X) -> c_6(cons#(X,n__from(s(X))))
            from#(X) -> c_7(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#} and constructors {n__cons,n__from,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: cons#(X1,X2) -> c_5(X1,X2)
          
        The strictly oriented rules are moved into the weak component.
*** Step 4.b:1.a:1: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(X1,X2) -> c_5(X1,X2)
        - Weak DPs:
            activate#(X) -> c_2(X)
            activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            from#(X) -> c_6(cons#(X,n__from(s(X))))
            from#(X) -> c_7(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#} and constructors {n__cons,n__from,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_4) = {1},
          uargs(c_6) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                p(2nd) = [0]          
           p(activate) = [0]          
               p(cons) = [2] x2 + [0] 
               p(from) = [4] x1 + [0] 
            p(n__cons) = [2]          
            p(n__from) = [0]          
                  p(s) = [0]          
               p(2nd#) = [1] x1 + [1] 
          p(activate#) = [8] x1 + [14]
              p(cons#) = [3]          
              p(from#) = [3]          
                p(c_1) = [1] x1 + [0] 
                p(c_2) = [8] x1 + [14]
                p(c_3) = [8] x1 + [6] 
                p(c_4) = [2] x1 + [8] 
                p(c_5) = [2]          
                p(c_6) = [1] x1 + [0] 
                p(c_7) = [0]          
        
        Following rules are strictly oriented:
        cons#(X1,X2) = [3]       
                     > [2]       
                     = c_5(X1,X2)
        
        
        Following rules are (at-least) weakly oriented:
                     activate#(X) =  [8] X + [14]               
                                  >= [8] X + [14]               
                                  =  c_2(X)                     
        
        activate#(n__cons(X1,X2)) =  [30]                       
                                  >= [30]                       
                                  =  c_3(cons#(X1,X2))          
        
            activate#(n__from(X)) =  [14]                       
                                  >= [14]                       
                                  =  c_4(from#(X))              
        
                         from#(X) =  [3]                        
                                  >= [3]                        
                                  =  c_6(cons#(X,n__from(s(X))))
        
                         from#(X) =  [3]                        
                                  >= [0]                        
                                  =  c_7(X)                     
        
*** Step 4.b:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(X) -> c_2(X)
            activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            cons#(X1,X2) -> c_5(X1,X2)
            from#(X) -> c_6(cons#(X,n__from(s(X))))
            from#(X) -> c_7(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#} and constructors {n__cons,n__from,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 4.b:1.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            activate#(X) -> c_2(X)
            activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
            activate#(n__from(X)) -> c_4(from#(X))
            cons#(X1,X2) -> c_5(X1,X2)
            from#(X) -> c_6(cons#(X,n__from(s(X))))
            from#(X) -> c_7(X)
        - Signature:
            {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#} and constructors {n__cons,n__from,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:activate#(X) -> c_2(X)
             -->_1 from#(X) -> c_7(X):6
             -->_1 from#(X) -> c_6(cons#(X,n__from(s(X)))):5
             -->_1 cons#(X1,X2) -> c_5(X1,X2):4
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):3
             -->_1 activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)):2
             -->_1 activate#(X) -> c_2(X):1
          
          2:W:activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
             -->_1 cons#(X1,X2) -> c_5(X1,X2):4
          
          3:W:activate#(n__from(X)) -> c_4(from#(X))
             -->_1 from#(X) -> c_7(X):6
             -->_1 from#(X) -> c_6(cons#(X,n__from(s(X)))):5
          
          4:W:cons#(X1,X2) -> c_5(X1,X2)
             -->_2 from#(X) -> c_7(X):6
             -->_1 from#(X) -> c_7(X):6
             -->_2 from#(X) -> c_6(cons#(X,n__from(s(X)))):5
             -->_1 from#(X) -> c_6(cons#(X,n__from(s(X)))):5
             -->_2 cons#(X1,X2) -> c_5(X1,X2):4
             -->_1 cons#(X1,X2) -> c_5(X1,X2):4
             -->_2 activate#(n__from(X)) -> c_4(from#(X)):3
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):3
             -->_2 activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)):2
             -->_1 activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)):2
             -->_2 activate#(X) -> c_2(X):1
             -->_1 activate#(X) -> c_2(X):1
          
          5:W:from#(X) -> c_6(cons#(X,n__from(s(X))))
             -->_1 cons#(X1,X2) -> c_5(X1,X2):4
          
          6:W:from#(X) -> c_7(X)
             -->_1 from#(X) -> c_7(X):6
             -->_1 from#(X) -> c_6(cons#(X,n__from(s(X)))):5
             -->_1 cons#(X1,X2) -> c_5(X1,X2):4
             -->_1 activate#(n__from(X)) -> c_4(from#(X)):3
             -->_1 activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2)):2
             -->_1 activate#(X) -> c_2(X):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: activate#(X) -> c_2(X)
          6: from#(X) -> c_7(X)
          4: cons#(X1,X2) -> c_5(X1,X2)
          5: from#(X) -> c_6(cons#(X,n__from(s(X))))
          3: activate#(n__from(X)) -> c_4(from#(X))
          2: activate#(n__cons(X1,X2)) -> c_3(cons#(X1,X2))
*** Step 4.b:1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {2nd/1,activate/1,cons/2,from/1,2nd#/1,activate#/1,cons#/2,from#/1} / {n__cons/2,n__from/1,s/1,c_1/1,c_2/1
            ,c_3/1,c_4/1,c_5/2,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {2nd#,activate#,cons#,from#} and constructors {n__cons,n__from,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))