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


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WORST_CASE(?,O(n^1))
* Step 1: WeightGap.  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:
            innermost 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) = [1]         
            p(a__g) = [1] x1 + [0]
            p(a__h) = [1] x1 + [0]
               p(c) = [0]         
               p(d) = [0]         
               p(g) = [1] x1 + [0]
               p(h) = [1] x1 + [0]
            p(mark) = [1] x1 + [5]
          
          Following rules are strictly oriented:
              a__c() = [1]        
                     > [0]        
                     = c()        
          
              a__c() = [1]        
                     > [0]        
                     = d()        
          
           mark(c()) = [5]        
                     > [1]        
                     = a__c()     
          
           mark(d()) = [5]        
                     > [0]        
                     = d()        
          
          mark(g(X)) = [1] X + [5]
                     > [1] X + [0]
                     = a__g(X)    
          
          mark(h(X)) = [1] X + [5]
                     > [1] X + [0]
                     = a__h(X)    
          
          
          Following rules are (at-least) weakly oriented:
            a__g(X) =  [1] X + [0]
                    >= [1] X + [0]
                    =  a__h(X)    
          
            a__g(X) =  [1] X + [0]
                    >= [1] X + [0]
                    =  g(X)       
          
            a__h(X) =  [1] X + [0]
                    >= [1] X + [0]
                    =  h(X)       
          
          a__h(d()) =  [0]        
                    >= [0]        
                    =  a__g(c())  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__g(X) -> a__h(X)
            a__g(X) -> g(X)
            a__h(X) -> h(X)
            a__h(d()) -> a__g(c())
        - Weak TRS:
            a__c() -> c()
            a__c() -> d()
            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:
            innermost 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) = [10]        
            p(a__g) = [1] x1 + [6]
            p(a__h) = [1] x1 + [9]
               p(c) = [10]        
               p(d) = [10]        
               p(g) = [1] x1 + [6]
               p(h) = [1] x1 + [9]
            p(mark) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          a__h(d()) = [19]     
                    > [16]     
                    = a__g(c())
          
          
          Following rules are (at-least) weakly oriented:
              a__c() =  [10]       
                     >= [10]       
                     =  c()        
          
              a__c() =  [10]       
                     >= [10]       
                     =  d()        
          
             a__g(X) =  [1] X + [6]
                     >= [1] X + [9]
                     =  a__h(X)    
          
             a__g(X) =  [1] X + [6]
                     >= [1] X + [6]
                     =  g(X)       
          
             a__h(X) =  [1] X + [9]
                     >= [1] X + [9]
                     =  h(X)       
          
           mark(c()) =  [10]       
                     >= [10]       
                     =  a__c()     
          
           mark(d()) =  [10]       
                     >= [10]       
                     =  d()        
          
          mark(g(X)) =  [1] X + [6]
                     >= [1] X + [6]
                     =  a__g(X)    
          
          mark(h(X)) =  [1] X + [9]
                     >= [1] X + [9]
                     =  a__h(X)    
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__g(X) -> a__h(X)
            a__g(X) -> g(X)
            a__h(X) -> h(X)
        - Weak TRS:
            a__c() -> c()
            a__c() -> d()
            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:
            innermost 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) = [10]         
          p(a__g) = [1] x1 + [2] 
          p(a__h) = [1] x1 + [0] 
             p(c) = [0]          
             p(d) = [7]          
             p(g) = [1] x1 + [2] 
             p(h) = [1] x1 + [0] 
          p(mark) = [1] x1 + [11]
        
        Following rules are strictly oriented:
        a__g(X) = [1] X + [2]
                > [1] X + [0]
                = a__h(X)    
        
        
        Following rules are (at-least) weakly oriented:
            a__c() =  [10]        
                   >= [0]         
                   =  c()         
        
            a__c() =  [10]        
                   >= [7]         
                   =  d()         
        
           a__g(X) =  [1] X + [2] 
                   >= [1] X + [2] 
                   =  g(X)        
        
           a__h(X) =  [1] X + [0] 
                   >= [1] X + [0] 
                   =  h(X)        
        
         a__h(d()) =  [7]         
                   >= [2]         
                   =  a__g(c())   
        
         mark(c()) =  [11]        
                   >= [10]        
                   =  a__c()      
        
         mark(d()) =  [18]        
                   >= [7]         
                   =  d()         
        
        mark(g(X)) =  [1] X + [13]
                   >= [1] X + [2] 
                   =  a__g(X)     
        
        mark(h(X)) =  [1] X + [11]
                   >= [1] X + [0] 
                   =  a__h(X)     
        
* Step 4: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__g(X) -> g(X)
            a__h(X) -> h(X)
        - Weak TRS:
            a__c() -> c()
            a__c() -> d()
            a__g(X) -> a__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:
            innermost 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) = [9]         
            p(a__g) = [1] x1 + [3]
            p(a__h) = [1] x1 + [3]
               p(c) = [9]         
               p(d) = [9]         
               p(g) = [1] x1 + [5]
               p(h) = [1] x1 + [2]
            p(mark) = [1] x1 + [2]
          
          Following rules are strictly oriented:
          a__h(X) = [1] X + [3]
                  > [1] X + [2]
                  = h(X)       
          
          
          Following rules are (at-least) weakly oriented:
              a__c() =  [9]        
                     >= [9]        
                     =  c()        
          
              a__c() =  [9]        
                     >= [9]        
                     =  d()        
          
             a__g(X) =  [1] X + [3]
                     >= [1] X + [3]
                     =  a__h(X)    
          
             a__g(X) =  [1] X + [3]
                     >= [1] X + [5]
                     =  g(X)       
          
           a__h(d()) =  [12]       
                     >= [12]       
                     =  a__g(c())  
          
           mark(c()) =  [11]       
                     >= [9]        
                     =  a__c()     
          
           mark(d()) =  [11]       
                     >= [9]        
                     =  d()        
          
          mark(g(X)) =  [1] X + [7]
                     >= [1] X + [3]
                     =  a__g(X)    
          
          mark(h(X)) =  [1] X + [4]
                     >= [1] X + [3]
                     =  a__h(X)    
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a__g(X) -> g(X)
        - Weak TRS:
            a__c() -> c()
            a__c() -> d()
            a__g(X) -> a__h(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:
            innermost 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) = [0]         
          p(a__g) = [1] x1 + [4]
          p(a__h) = [1] x1 + [4]
             p(c) = [0]         
             p(d) = [0]         
             p(g) = [1] x1 + [0]
             p(h) = [1] x1 + [4]
          p(mark) = [1] x1 + [8]
        
        Following rules are strictly oriented:
        a__g(X) = [1] X + [4]
                > [1] X + [0]
                = g(X)       
        
        
        Following rules are (at-least) weakly oriented:
            a__c() =  [0]         
                   >= [0]         
                   =  c()         
        
            a__c() =  [0]         
                   >= [0]         
                   =  d()         
        
           a__g(X) =  [1] X + [4] 
                   >= [1] X + [4] 
                   =  a__h(X)     
        
           a__h(X) =  [1] X + [4] 
                   >= [1] X + [4] 
                   =  h(X)        
        
         a__h(d()) =  [4]         
                   >= [4]         
                   =  a__g(c())   
        
         mark(c()) =  [8]         
                   >= [0]         
                   =  a__c()      
        
         mark(d()) =  [8]         
                   >= [0]         
                   =  d()         
        
        mark(g(X)) =  [1] X + [8] 
                   >= [1] X + [4] 
                   =  a__g(X)     
        
        mark(h(X)) =  [1] X + [12]
                   >= [1] X + [4] 
                   =  a__h(X)     
        
* Step 6: 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:
            innermost 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))