/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2} / {++/2,nil/0,u/0,v/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {merge} and constructors {++,nil,u,v}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2} / {++/2,nil/0,u/0,v/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {merge} and constructors {++,nil,u,v}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2} / {++/2,nil/0,u/0,v/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {merge} and constructors {++,nil,u,v}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          merge(y,++(u(),v())){y -> ++(x,y)} =
            merge(++(x,y),++(u(),v())) ->^+ ++(x,merge(y,++(u(),v())))
              = C[merge(y,++(u(),v())) = merge(y,++(u(),v())){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2} / {++/2,nil/0,u/0,v/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {merge} and constructors {++,nil,u,v}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          merge#(x,nil()) -> c_1()
          merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v())))
          merge#(++(x,y),++(u(),v())) -> c_3(merge#(++(x,y),v()))
          merge#(nil(),y) -> c_4()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            merge#(x,nil()) -> c_1()
            merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v())))
            merge#(++(x,y),++(u(),v())) -> c_3(merge#(++(x,y),v()))
            merge#(nil(),y) -> c_4()
        - Weak TRS:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2,merge#/2} / {++/2,nil/0,u/0,v/0,c_1/0,c_2/1,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {merge#} and constructors {++,nil,u,v}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4}
        by application of
          Pre({1,3,4}) = {2}.
        Here rules are labelled as follows:
          1: merge#(x,nil()) -> c_1()
          2: merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v())))
          3: merge#(++(x,y),++(u(),v())) -> c_3(merge#(++(x,y),v()))
          4: merge#(nil(),y) -> c_4()
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v())))
        - Weak DPs:
            merge#(x,nil()) -> c_1()
            merge#(++(x,y),++(u(),v())) -> c_3(merge#(++(x,y),v()))
            merge#(nil(),y) -> c_4()
        - Weak TRS:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2,merge#/2} / {++/2,nil/0,u/0,v/0,c_1/0,c_2/1,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {merge#} and constructors {++,nil,u,v}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v())))
             -->_1 merge#(nil(),y) -> c_4():4
             -->_1 merge#(++(x,y),++(u(),v())) -> c_3(merge#(++(x,y),v())):3
             -->_1 merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v()))):1
          
          2:W:merge#(x,nil()) -> c_1()
             
          
          3:W:merge#(++(x,y),++(u(),v())) -> c_3(merge#(++(x,y),v()))
             
          
          4:W:merge#(nil(),y) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: merge#(x,nil()) -> c_1()
          3: merge#(++(x,y),++(u(),v())) -> c_3(merge#(++(x,y),v()))
          4: merge#(nil(),y) -> c_4()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v())))
        - Weak TRS:
            merge(x,nil()) -> x
            merge(++(x,y),++(u(),v())) -> ++(x,merge(y,++(u(),v())))
            merge(++(x,y),++(u(),v())) -> ++(u(),merge(++(x,y),v()))
            merge(nil(),y) -> y
        - Signature:
            {merge/2,merge#/2} / {++/2,nil/0,u/0,v/0,c_1/0,c_2/1,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {merge#} and constructors {++,nil,u,v}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v())))
** Step 1.b:5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v())))
        - Signature:
            {merge/2,merge#/2} / {++/2,nil/0,u/0,v/0,c_1/0,c_2/1,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {merge#} and constructors {++,nil,u,v}
    + 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_2) = {1}
        
        Following symbols are considered usable:
          {merge#}
        TcT has computed the following interpretation:
              p(++) = [1] x1 + [1] x2 + [1]
           p(merge) = [1] x1 + [0]         
             p(nil) = [1]                  
               p(u) = [5]                  
               p(v) = [2]                  
          p(merge#) = [4] x1 + [0]         
             p(c_1) = [2]                  
             p(c_2) = [1] x1 + [0]         
             p(c_3) = [1] x1 + [0]         
             p(c_4) = [0]                  
        
        Following rules are strictly oriented:
        merge#(++(x,y),++(u(),v())) = [4] x + [4] y + [4]       
                                    > [4] y + [0]               
                                    = c_2(merge#(y,++(u(),v())))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:6: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            merge#(++(x,y),++(u(),v())) -> c_2(merge#(y,++(u(),v())))
        - Signature:
            {merge/2,merge#/2} / {++/2,nil/0,u/0,v/0,c_1/0,c_2/1,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {merge#} and constructors {++,nil,u,v}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))