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


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WORST_CASE(?,O(n^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            max(L(x)) -> x
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        - Signature:
            {max/1} / {0/0,L/1,N/2,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,L,N,max,s}
    + 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(0) = [8]                  
              p(L) = [1] x1 + [8]         
              p(N) = [1] x1 + [1] x2 + [0]
            p(max) = [1] x1 + [1]         
              p(s) = [1] x1 + [3]         
          
          Following rules are strictly oriented:
                        max(L(x)) = [1] x + [9]         
                                  > [1] x + [0]         
                                  = x                   
          
              max(N(L(0()),L(y))) = [1] y + [25]        
                                  > [1] y + [0]         
                                  = y                   
          
          max(N(L(s(x)),L(s(y)))) = [1] x + [1] y + [23]
                                  > [1] x + [1] y + [20]
                                  = s(max(N(L(x),L(y))))
          
          
          Following rules are (at-least) weakly oriented:
          max(N(L(x),N(y,z))) =  [1] x + [1] y + [1] z + [9] 
                              >= [1] x + [1] y + [1] z + [18]
                              =  max(N(L(x),L(max(N(y,z))))) 
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: Bounds.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
        - Weak TRS:
            max(L(x)) -> x
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        - Signature:
            {max/1} / {0/0,L/1,N/2,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,L,N,max,s}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 1.
        The enriched problem is compatible with follwoing automaton.
          0_0() -> 1
          0_0() -> 5
          0_0() -> 7
          L_0(1) -> 1
          L_0(1) -> 5
          L_0(1) -> 7
          L_1(1) -> 3
          L_1(5) -> 4
          L_1(7) -> 3
          N_0(1,1) -> 1
          N_0(1,1) -> 5
          N_0(1,1) -> 7
          N_1(1,1) -> 6
          N_1(3,3) -> 8
          N_1(3,4) -> 2
          max_0(1) -> 1
          max_0(1) -> 5
          max_0(1) -> 7
          max_1(2) -> 1
          max_1(2) -> 5
          max_1(2) -> 7
          max_1(6) -> 1
          max_1(6) -> 5
          max_1(6) -> 7
          max_1(8) -> 7
          s_0(1) -> 1
          s_0(1) -> 5
          s_0(1) -> 7
          s_1(7) -> 1
          s_1(7) -> 5
          s_1(7) -> 7
          1 -> 5
          1 -> 7
          5 -> 1
          5 -> 7
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            max(L(x)) -> x
            max(N(L(x),N(y,z))) -> max(N(L(x),L(max(N(y,z)))))
            max(N(L(0()),L(y))) -> y
            max(N(L(s(x)),L(s(y)))) -> s(max(N(L(x),L(y))))
        - Signature:
            {max/1} / {0/0,L/1,N/2,s/1}
        - Obligation:
             derivational complexity wrt. signature {0,L,N,max,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))