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


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WORST_CASE(?,O(n^1))
* Step 1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            c() -> d()
            f(f(X)) -> c()
            h(X) -> c()
        - Signature:
            {c/0,f/1,h/1} / {d/0}
        - Obligation:
            innermost derivational complexity wrt. signature {c,d,f,h}
    + 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(c) = [8]         
          p(d) = [0]         
          p(f) = [1] x1 + [4]
          p(h) = [1] x1 + [8]
        
        Following rules are strictly oriented:
        c() = [8]
            > [0]
            = d()
        
        
        Following rules are (at-least) weakly oriented:
        f(f(X)) =  [1] X + [8]
                >= [8]        
                =  c()        
        
           h(X) =  [1] X + [8]
                >= [8]        
                =  c()        
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(f(X)) -> c()
            h(X) -> c()
        - Weak TRS:
            c() -> d()
        - Signature:
            {c/0,f/1,h/1} / {d/0}
        - Obligation:
            innermost derivational complexity wrt. signature {c,d,f,h}
    + 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(c) = [0]         
          p(d) = [0]         
          p(f) = [1] x1 + [4]
          p(h) = [1] x1 + [0]
        
        Following rules are strictly oriented:
        f(f(X)) = [1] X + [8]
                > [0]        
                = c()        
        
        
        Following rules are (at-least) weakly oriented:
         c() =  [0]        
             >= [0]        
             =  d()        
        
        h(X) =  [1] X + [0]
             >= [0]        
             =  c()        
        
* Step 3: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            h(X) -> c()
        - Weak TRS:
            c() -> d()
            f(f(X)) -> c()
        - Signature:
            {c/0,f/1,h/1} / {d/0}
        - Obligation:
            innermost derivational complexity wrt. signature {c,d,f,h}
    + 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(c) = [4]          
          p(d) = [0]          
          p(f) = [1] x1 + [2] 
          p(h) = [1] x1 + [12]
        
        Following rules are strictly oriented:
        h(X) = [1] X + [12]
             > [4]         
             = c()         
        
        
        Following rules are (at-least) weakly oriented:
            c() =  [4]        
                >= [0]        
                =  d()        
        
        f(f(X)) =  [1] X + [4]
                >= [4]        
                =  c()        
        
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            c() -> d()
            f(f(X)) -> c()
            h(X) -> c()
        - Signature:
            {c/0,f/1,h/1} / {d/0}
        - Obligation:
            innermost derivational complexity wrt. signature {c,d,f,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))