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


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WORST_CASE(?,O(n^1))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__c() -> c()
            a__c() -> d()
            a__g(X) -> a__h(X)
            a__g(X) -> g(X)
            a__h(X) -> h(X)
            a__h(d()) -> a__g(c())
            mark(c()) -> a__c()
            mark(d()) -> d()
            mark(g(X)) -> a__g(X)
            mark(h(X)) -> a__h(X)
        - Signature:
            {a__c/0,a__g/1,a__h/1,mark/1} / {c/0,d/0,g/1,h/1}
        - Obligation:
             derivational complexity wrt. signature {a__c,a__g,a__h,c,d,g,h,mark}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a__c) = [14]         
          p(a__g) = [1] x1 + [1] 
          p(a__h) = [1] x1 + [0] 
             p(c) = [2]          
             p(d) = [3]          
             p(g) = [1] x1 + [1] 
             p(h) = [1] x1 + [0] 
          p(mark) = [1] x1 + [14]
        
        Following rules are strictly oriented:
            a__c() = [14]        
                   > [2]         
                   = c()         
        
            a__c() = [14]        
                   > [3]         
                   = d()         
        
           a__g(X) = [1] X + [1] 
                   > [1] X + [0] 
                   = a__h(X)     
        
         mark(c()) = [16]        
                   > [14]        
                   = a__c()      
        
         mark(d()) = [17]        
                   > [3]         
                   = d()         
        
        mark(g(X)) = [1] X + [15]
                   > [1] X + [1] 
                   = a__g(X)     
        
        mark(h(X)) = [1] X + [14]
                   > [1] X + [0] 
                   = a__h(X)     
        
        
        Following rules are (at-least) weakly oriented:
          a__g(X) =  [1] X + [1]
                  >= [1] X + [1]
                  =  g(X)       
        
          a__h(X) =  [1] X + [0]
                  >= [1] X + [0]
                  =  h(X)       
        
        a__h(d()) =  [3]        
                  >= [3]        
                  =  a__g(c())  
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__g(X) -> g(X)
            a__h(X) -> h(X)
            a__h(d()) -> a__g(c())
        - Weak TRS:
            a__c() -> c()
            a__c() -> d()
            a__g(X) -> a__h(X)
            mark(c()) -> a__c()
            mark(d()) -> d()
            mark(g(X)) -> a__g(X)
            mark(h(X)) -> a__h(X)
        - Signature:
            {a__c/0,a__g/1,a__h/1,mark/1} / {c/0,d/0,g/1,h/1}
        - Obligation:
             derivational complexity wrt. signature {a__c,a__g,a__h,c,d,g,h,mark}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation:
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a__c) = [1]         
          p(a__g) = [1] x1 + [1]
          p(a__h) = [1] x1 + [1]
             p(c) = [0]         
             p(d) = [0]         
             p(g) = [1] x1 + [0]
             p(h) = [1] x1 + [0]
          p(mark) = [1] x1 + [8]
        
        Following rules are strictly oriented:
        a__g(X) = [1] X + [1]
                > [1] X + [0]
                = g(X)       
        
        a__h(X) = [1] X + [1]
                > [1] X + [0]
                = h(X)       
        
        
        Following rules are (at-least) weakly oriented:
            a__c() =  [1]        
                   >= [0]        
                   =  c()        
        
            a__c() =  [1]        
                   >= [0]        
                   =  d()        
        
           a__g(X) =  [1] X + [1]
                   >= [1] X + [1]
                   =  a__h(X)    
        
         a__h(d()) =  [1]        
                   >= [1]        
                   =  a__g(c())  
        
         mark(c()) =  [8]        
                   >= [1]        
                   =  a__c()     
        
         mark(d()) =  [8]        
                   >= [0]        
                   =  d()        
        
        mark(g(X)) =  [1] X + [8]
                   >= [1] X + [1]
                   =  a__g(X)    
        
        mark(h(X)) =  [1] X + [8]
                   >= [1] X + [1]
                   =  a__h(X)    
        
* Step 3: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__h(d()) -> a__g(c())
        - Weak TRS:
            a__c() -> c()
            a__c() -> d()
            a__g(X) -> a__h(X)
            a__g(X) -> g(X)
            a__h(X) -> h(X)
            mark(c()) -> a__c()
            mark(d()) -> d()
            mark(g(X)) -> a__g(X)
            mark(h(X)) -> a__h(X)
        - Signature:
            {a__c/0,a__g/1,a__h/1,mark/1} / {c/0,d/0,g/1,h/1}
        - Obligation:
             derivational complexity wrt. signature {a__c,a__g,a__h,c,d,g,h,mark}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind triangular matrix interpretation:
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
            p(a__c) = [8]          
            p(a__g) = [1] x1 + [8] 
            p(a__h) = [1] x1 + [8] 
               p(c) = [0]          
               p(d) = [1]          
               p(g) = [1] x1 + [1] 
               p(h) = [1] x1 + [8] 
            p(mark) = [1] x1 + [12]
          
          Following rules are strictly oriented:
          a__h(d()) = [9]      
                    > [8]      
                    = a__g(c())
          
          
          Following rules are (at-least) weakly oriented:
              a__c() =  [8]         
                     >= [0]         
                     =  c()         
          
              a__c() =  [8]         
                     >= [1]         
                     =  d()         
          
             a__g(X) =  [1] X + [8] 
                     >= [1] X + [8] 
                     =  a__h(X)     
          
             a__g(X) =  [1] X + [8] 
                     >= [1] X + [1] 
                     =  g(X)        
          
             a__h(X) =  [1] X + [8] 
                     >= [1] X + [8] 
                     =  h(X)        
          
           mark(c()) =  [12]        
                     >= [8]         
                     =  a__c()      
          
           mark(d()) =  [13]        
                     >= [1]         
                     =  d()         
          
          mark(g(X)) =  [1] X + [13]
                     >= [1] X + [8] 
                     =  a__g(X)     
          
          mark(h(X)) =  [1] X + [20]
                     >= [1] X + [8] 
                     =  a__h(X)     
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a__c() -> c()
            a__c() -> d()
            a__g(X) -> a__h(X)
            a__g(X) -> g(X)
            a__h(X) -> h(X)
            a__h(d()) -> a__g(c())
            mark(c()) -> a__c()
            mark(d()) -> d()
            mark(g(X)) -> a__g(X)
            mark(h(X)) -> a__h(X)
        - Signature:
            {a__c/0,a__g/1,a__h/1,mark/1} / {c/0,d/0,g/1,h/1}
        - Obligation:
             derivational complexity wrt. signature {a__c,a__g,a__h,c,d,g,h,mark}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))