/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:
            f(g(h(a(),b()),c()),d()) -> if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
            f(g(i(a(),b(),b'()),c()),d()) -> if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
        - Signature:
            {f/2} / {./2,a/0,b/0,b'/0,c/0,d/0,d'/0,e/0,g/2,h/2,i/3,if/3}
        - Obligation:
            innermost derivational complexity wrt. signature {.,a,b,b',c,d,d',e,f,g,h,i,if}
    + 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(.) = [1] x1 + [1] x2 + [0]         
             p(a) = [15]                          
             p(b) = [8]                           
            p(b') = [8]                           
             p(c) = [0]                           
             p(d) = [0]                           
            p(d') = [0]                           
             p(e) = [0]                           
             p(f) = [1] x1 + [1] x2 + [0]         
             p(g) = [1] x1 + [1] x2 + [0]         
             p(h) = [1] x1 + [1] x2 + [0]         
             p(i) = [1] x1 + [1] x2 + [1] x3 + [0]
            p(if) = [1] x1 + [1] x2 + [1] x3 + [0]
          
          Following rules are strictly oriented:
          f(g(i(a(),b(),b'()),c()),d()) = [31]                                          
                                        > [16]                                          
                                        = if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
          
          
          Following rules are (at-least) weakly oriented:
          f(g(h(a(),b()),c()),d()) =  [23]                                               
                                   >= [31]                                               
                                   =  if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(g(h(a(),b()),c()),d()) -> if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
        - Weak TRS:
            f(g(i(a(),b(),b'()),c()),d()) -> if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
        - Signature:
            {f/2} / {./2,a/0,b/0,b'/0,c/0,d/0,d'/0,e/0,g/2,h/2,i/3,if/3}
        - Obligation:
            innermost derivational complexity wrt. signature {.,a,b,b',c,d,d',e,f,g,h,i,if}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
           p(.) = [1 1] x1 + [1 0] x2 + [0]           
                  [0 0]      [0 0]      [0]           
           p(a) = [0]                                 
                  [1]                                 
           p(b) = [0]                                 
                  [0]                                 
          p(b') = [2]                                 
                  [2]                                 
           p(c) = [0]                                 
                  [0]                                 
           p(d) = [1]                                 
                  [0]                                 
          p(d') = [0]                                 
                  [0]                                 
           p(e) = [0]                                 
                  [4]                                 
           p(f) = [1 1] x1 + [1 1] x2 + [0]           
                  [0 0]      [0 0]      [0]           
           p(g) = [1 1] x1 + [1 2] x2 + [0]           
                  [0 0]      [0 0]      [1]           
           p(h) = [1 7] x1 + [1 0] x2 + [0]           
                  [0 0]      [0 0]      [1]           
           p(i) = [1 0] x1 + [1 1] x2 + [1 2] x3 + [1]
                  [0 0]      [0 0]      [0 0]      [0]
          p(if) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [0]
                  [0 0]      [0 0]      [0 0]      [0]
        
        Following rules are strictly oriented:
        f(g(h(a(),b()),c()),d()) = [10]                                               
                                   [0]                                                
                                 > [9]                                                
                                   [0]                                                
                                 = if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
        
        
        Following rules are (at-least) weakly oriented:
        f(g(i(a(),b(),b'()),c()),d()) =  [9]                                           
                                         [0]                                           
                                      >= [4]                                           
                                         [0]                                           
                                      =  if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(g(h(a(),b()),c()),d()) -> if(e(),f(.(b(),g(h(a(),b()),c())),d()),f(c(),d'()))
            f(g(i(a(),b(),b'()),c()),d()) -> if(e(),f(.(b(),c()),d'()),f(.(b'(),c()),d'()))
        - Signature:
            {f/2} / {./2,a/0,b/0,b'/0,c/0,d/0,d'/0,e/0,g/2,h/2,i/3,if/3}
        - Obligation:
            innermost derivational complexity wrt. signature {.,a,b,b',c,d,d',e,f,g,h,i,if}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))