/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:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3} / {a/0,b/0,c/1,d/0,k/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g,h,u} and constructors {a,b,c,d,k}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3} / {a/0,b/0,c/1,d/0,k/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g,h,u} and constructors {a,b,c,d,k}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
          g#(X) -> c_2(u#(h(X),h(X),X))
          h#(d()) -> c_3()
          h#(d()) -> c_4()
          u#(d(),c(Y),X) -> c_5(Y)
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            g#(X) -> c_2(u#(h(X),h(X),X))
            h#(d()) -> c_3()
            h#(d()) -> c_4()
            u#(d(),c(Y),X) -> c_5(Y)
        - Strict TRS:
            f(k(a()),k(b()),X) -> f(X,X,X)
            g(X) -> u(h(X),h(X),X)
            h(d()) -> c(a())
            h(d()) -> c(b())
            u(d(),c(Y),X) -> k(Y)
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          h(d()) -> c(a())
          h(d()) -> c(b())
          f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
          g#(X) -> c_2(u#(h(X),h(X),X))
          h#(d()) -> c_3()
          h#(d()) -> c_4()
          u#(d(),c(Y),X) -> c_5(Y)
* Step 4: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            g#(X) -> c_2(u#(h(X),h(X),X))
            h#(d()) -> c_3()
            h#(d()) -> c_4()
            u#(d(),c(Y),X) -> c_5(Y)
        - Strict TRS:
            h(d()) -> c(a())
            h(d()) -> c(b())
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + 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(u#) = {1,2},
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(a) = [0]                  
              p(b) = [0]                  
              p(c) = [0]                  
              p(d) = [0]                  
              p(f) = [0]                  
              p(g) = [0]                  
              p(h) = [5]                  
              p(k) = [1] x1 + [0]         
              p(u) = [0]                  
             p(f#) = [0]                  
             p(g#) = [0]                  
             p(h#) = [0]                  
             p(u#) = [1] x1 + [1] x2 + [0]
            p(c_1) = [8] x1 + [0]         
            p(c_2) = [1] x1 + [0]         
            p(c_3) = [0]                  
            p(c_4) = [0]                  
            p(c_5) = [0]                  
          
          Following rules are strictly oriented:
          h(d()) = [5]   
                 > [0]   
                 = c(a())
          
          h(d()) = [5]   
                 > [0]   
                 = c(b())
          
          
          Following rules are (at-least) weakly oriented:
          f#(k(a()),k(b()),X) =  [0]                 
                              >= [0]                 
                              =  c_1(f#(X,X,X))      
          
                        g#(X) =  [0]                 
                              >= [10]                
                              =  c_2(u#(h(X),h(X),X))
          
                      h#(d()) =  [0]                 
                              >= [0]                 
                              =  c_3()               
          
                      h#(d()) =  [0]                 
                              >= [0]                 
                              =  c_4()               
          
               u#(d(),c(Y),X) =  [0]                 
                              >= [0]                 
                              =  c_5(Y)              
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            g#(X) -> c_2(u#(h(X),h(X),X))
            h#(d()) -> c_3()
            h#(d()) -> c_4()
            u#(d(),c(Y),X) -> c_5(Y)
        - Weak TRS:
            h(d()) -> c(a())
            h(d()) -> c(b())
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4}
        by application of
          Pre({3,4}) = {5}.
        Here rules are labelled as follows:
          1: f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
          2: g#(X) -> c_2(u#(h(X),h(X),X))
          3: h#(d()) -> c_3()
          4: h#(d()) -> c_4()
          5: u#(d(),c(Y),X) -> c_5(Y)
* Step 6: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            g#(X) -> c_2(u#(h(X),h(X),X))
            u#(d(),c(Y),X) -> c_5(Y)
        - Weak DPs:
            h#(d()) -> c_3()
            h#(d()) -> c_4()
        - Weak TRS:
            h(d()) -> c(a())
            h(d()) -> c(b())
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
             -->_1 f#(k(a()),k(b()),X) -> c_1(f#(X,X,X)):1
          
          2:S:g#(X) -> c_2(u#(h(X),h(X),X))
             -->_1 u#(d(),c(Y),X) -> c_5(Y):3
          
          3:S:u#(d(),c(Y),X) -> c_5(Y)
             -->_1 h#(d()) -> c_4():5
             -->_1 h#(d()) -> c_3():4
             -->_1 u#(d(),c(Y),X) -> c_5(Y):3
             -->_1 g#(X) -> c_2(u#(h(X),h(X),X)):2
             -->_1 f#(k(a()),k(b()),X) -> c_1(f#(X,X,X)):1
          
          4:W:h#(d()) -> c_3()
             
          
          5:W:h#(d()) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: h#(d()) -> c_3()
          5: h#(d()) -> c_4()
* Step 7: PredecessorEstimationCP.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            g#(X) -> c_2(u#(h(X),h(X),X))
            u#(d(),c(Y),X) -> c_5(Y)
        - Weak TRS:
            h(d()) -> c(a())
            h(d()) -> c(b())
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + 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: f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
          3: u#(d(),c(Y),X) -> c_5(Y)
          
        Consider the set of all dependency pairs
          1: f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
          2: g#(X) -> c_2(u#(h(X),h(X),X))
          3: u#(d(),c(Y),X) -> c_5(Y)
        Processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(1))
        BEST_CASE TIME (?,?)
        SPACE(?,?)on application of the dependency pairs
          {1,3}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,3}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
** Step 7.a:1: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            g#(X) -> c_2(u#(h(X),h(X),X))
            u#(d(),c(Y),X) -> c_5(Y)
        - Weak TRS:
            h(d()) -> c(a())
            h(d()) -> c(b())
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + 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_2) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(a) = [0]                           
            p(b) = [0]                           
            p(c) = [0]                           
            p(d) = [1]                           
            p(f) = [1] x1 + [1] x2 + [8] x3 + [1]
            p(g) = [1]                           
            p(h) = [0]                           
            p(k) = [2]                           
            p(u) = [1] x1 + [2] x2 + [8] x3 + [8]
           p(f#) = [14] x1 + [0]                 
           p(g#) = [1] x1 + [15]                 
           p(h#) = [0]                           
           p(u#) = [4] x1 + [1]                  
          p(c_1) = [0]                           
          p(c_2) = [8] x1 + [7]                  
          p(c_3) = [2]                           
          p(c_4) = [2]                           
          p(c_5) = [1]                           
        
        Following rules are strictly oriented:
        f#(k(a()),k(b()),X) = [28]          
                            > [0]           
                            = c_1(f#(X,X,X))
        
             u#(d(),c(Y),X) = [5]           
                            > [1]           
                            = c_5(Y)        
        
        
        Following rules are (at-least) weakly oriented:
         g#(X) =  [1] X + [15]        
               >= [15]                
               =  c_2(u#(h(X),h(X),X))
        
        h(d()) =  [0]                 
               >= [0]                 
               =  c(a())              
        
        h(d()) =  [0]                 
               >= [0]                 
               =  c(b())              
        
** Step 7.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(X) -> c_2(u#(h(X),h(X),X))
        - Weak DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            u#(d(),c(Y),X) -> c_5(Y)
        - Weak TRS:
            h(d()) -> c(a())
            h(d()) -> c(b())
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

** Step 7.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
            g#(X) -> c_2(u#(h(X),h(X),X))
            u#(d(),c(Y),X) -> c_5(Y)
        - Weak TRS:
            h(d()) -> c(a())
            h(d()) -> c(b())
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
             -->_1 f#(k(a()),k(b()),X) -> c_1(f#(X,X,X)):1
          
          2:W:g#(X) -> c_2(u#(h(X),h(X),X))
             -->_1 u#(d(),c(Y),X) -> c_5(Y):3
          
          3:W:u#(d(),c(Y),X) -> c_5(Y)
             -->_1 u#(d(),c(Y),X) -> c_5(Y):3
             -->_1 g#(X) -> c_2(u#(h(X),h(X),X)):2
             -->_1 f#(k(a()),k(b()),X) -> c_1(f#(X,X,X)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: g#(X) -> c_2(u#(h(X),h(X),X))
          3: u#(d(),c(Y),X) -> c_5(Y)
          1: f#(k(a()),k(b()),X) -> c_1(f#(X,X,X))
** Step 7.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            h(d()) -> c(a())
            h(d()) -> c(b())
        - Signature:
            {f/3,g/1,h/1,u/3,f#/3,g#/1,h#/1,u#/3} / {a/0,b/0,c/1,d/0,k/1,c_1/1,c_2/1,c_3/0,c_4/0,c_5/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#,h#,u#} and constructors {a,b,c,d,k}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))