/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: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            c(y) -> y
            c(a(a(0(),x),y)) -> a(c(c(c(0()))),y)
            c(c(c(y))) -> c(c(a(y,0())))
        - Signature:
            {c/1} / {0/0,a/2}
        - Obligation:
             derivational complexity wrt. signature {0,a,c}
    + 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) = [1]                  
            p(a) = [1] x1 + [1] x2 + [1]
            p(c) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          c(a(a(0(),x),y)) = [1] x + [1] y + [3]
                           > [1] y + [2]        
                           = a(c(c(c(0()))),y)  
          
          
          Following rules are (at-least) weakly oriented:
                c(y) =  [1] y + [0]   
                     >= [1] y + [0]   
                     =  y             
          
          c(c(c(y))) =  [1] y + [0]   
                     >= [1] y + [2]   
                     =  c(c(a(y,0())))
          
        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:
            c(y) -> y
            c(c(c(y))) -> c(c(a(y,0())))
        - Weak TRS:
            c(a(a(0(),x),y)) -> a(c(c(c(0()))),y)
        - Signature:
            {c/1} / {0/0,a/2}
        - Obligation:
             derivational complexity wrt. signature {0,a,c}
    + 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) = [4]                   
            p(a) = [1] x1 + [1] x2 + [10]
            p(c) = [1] x1 + [5]          
          
          Following rules are strictly oriented:
          c(y) = [1] y + [5]
               > [1] y + [0]
               = y          
          
          
          Following rules are (at-least) weakly oriented:
          c(a(a(0(),x),y)) =  [1] x + [1] y + [29]
                           >= [1] y + [29]        
                           =  a(c(c(c(0()))),y)   
          
                c(c(c(y))) =  [1] y + [15]        
                           >= [1] y + [24]        
                           =  c(c(a(y,0())))      
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: Bounds.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            c(c(c(y))) -> c(c(a(y,0())))
        - Weak TRS:
            c(y) -> y
            c(a(a(0(),x),y)) -> a(c(c(c(0()))),y)
        - Signature:
            {c/1} / {0/0,a/2}
        - Obligation:
             derivational complexity wrt. signature {0,a,c}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 3.
        The enriched problem is compatible with follwoing automaton.
          0_0() -> 1
          0_1() -> 4
          0_1() -> 5
          0_1() -> 6
          0_1() -> 7
          0_2() -> 12
          0_2() -> 13
          0_2() -> 14
          0_2() -> 15
          0_3() -> 18
          a_0(1,1) -> 1
          a_1(1,4) -> 1
          a_1(1,4) -> 2
          a_1(1,4) -> 3
          a_1(2,4) -> 1
          a_1(2,4) -> 2
          a_1(2,4) -> 3
          a_1(3,4) -> 1
          a_1(3,4) -> 2
          a_1(3,4) -> 3
          a_1(5,1) -> 1
          a_1(5,4) -> 1
          a_1(5,4) -> 2
          a_2(4,12) -> 5
          a_2(4,12) -> 10
          a_2(4,12) -> 11
          a_2(13,4) -> 1
          a_2(13,4) -> 2
          a_3(12,18) -> 13
          a_3(12,18) -> 16
          a_3(12,18) -> 17
          c_0(1) -> 1
          c_1(2) -> 1
          c_1(3) -> 1
          c_1(3) -> 2
          c_1(4) -> 5
          c_1(4) -> 6
          c_1(4) -> 7
          c_1(6) -> 5
          c_1(7) -> 5
          c_1(7) -> 6
          c_2(10) -> 5
          c_2(11) -> 5
          c_2(11) -> 10
          c_2(12) -> 13
          c_2(12) -> 14
          c_2(12) -> 15
          c_2(14) -> 13
          c_2(15) -> 13
          c_2(15) -> 14
          c_3(16) -> 13
          c_3(17) -> 13
          c_3(17) -> 16
          2 -> 1
          3 -> 1
          3 -> 2
          4 -> 5
          4 -> 6
          4 -> 7
          6 -> 5
          7 -> 5
          7 -> 6
          10 -> 5
          11 -> 5
          11 -> 10
          12 -> 13
          12 -> 14
          12 -> 15
          14 -> 13
          15 -> 13
          15 -> 14
          16 -> 13
          17 -> 13
          17 -> 16
* Step 4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            c(y) -> y
            c(a(a(0(),x),y)) -> a(c(c(c(0()))),y)
            c(c(c(y))) -> c(c(a(y,0())))
        - Signature:
            {c/1} / {0/0,a/2}
        - Obligation:
             derivational complexity wrt. signature {0,a,c}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))