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


--------------------------------------------------------------------------------


WORST_CASE(?,O(n^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1} / {a/0,b/0,c/1,d/1,g/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,e,f,g}
    + 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) = [0]          
            p(b) = [0]          
            p(c) = [1] x1 + [0] 
            p(d) = [1] x1 + [11]
            p(e) = [1] x1 + [3] 
            p(f) = [1] x1 + [0] 
            p(g) = [1] x1 + [0] 
          
          Following rules are strictly oriented:
          f(d(X)) = [1] X + [11]
                  > [1] X + [0] 
                  = X           
          
          
          Following rules are (at-least) weakly oriented:
            e(g(X)) =  [1] X + [3]
                    >= [1] X + [3]
                    =  e(X)       
          
             f(a()) =  [0]        
                    >= [0]        
                    =  f(c(a()))  
          
             f(a()) =  [0]        
                    >= [11]       
                    =  f(d(a()))  
          
            f(c(X)) =  [1] X + [0]
                    >= [1] X + [0]
                    =  X          
          
          f(c(a())) =  [0]        
                    >= [11]       
                    =  f(d(b()))  
          
          f(c(b())) =  [0]        
                    >= [11]       
                    =  f(d(a()))  
          
        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:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
        - Weak TRS:
            f(d(X)) -> X
        - Signature:
            {e/1,f/1} / {a/0,b/0,c/1,d/1,g/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,e,f,g}
    + 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) = [0]         
            p(b) = [13]        
            p(c) = [1] x1 + [0]
            p(d) = [1] x1 + [0]
            p(e) = [1] x1 + [3]
            p(f) = [1] x1 + [0]
            p(g) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          f(c(b())) = [13]     
                    > [0]      
                    = f(d(a()))
          
          
          Following rules are (at-least) weakly oriented:
            e(g(X)) =  [1] X + [3]
                    >= [1] X + [3]
                    =  e(X)       
          
             f(a()) =  [0]        
                    >= [0]        
                    =  f(c(a()))  
          
             f(a()) =  [0]        
                    >= [0]        
                    =  f(d(a()))  
          
            f(c(X)) =  [1] X + [0]
                    >= [1] X + [0]
                    =  X          
          
          f(c(a())) =  [0]        
                    >= [13]       
                    =  f(d(b()))  
          
            f(d(X)) =  [1] X + [0]
                    >= [1] X + [0]
                    =  X          
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
        - Weak TRS:
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1} / {a/0,b/0,c/1,d/1,g/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,e,f,g}
    + 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) = [1]         
            p(b) = [0]         
            p(c) = [1] x1 + [2]
            p(d) = [1] x1 + [0]
            p(e) = [1] x1 + [9]
            p(f) = [1] x1 + [8]
            p(g) = [1] x1 + [1]
          
          Following rules are strictly oriented:
            e(g(X)) = [1] X + [10]
                    > [1] X + [9] 
                    = e(X)        
          
            f(c(X)) = [1] X + [10]
                    > [1] X + [0] 
                    = X           
          
          f(c(a())) = [11]        
                    > [8]         
                    = f(d(b()))   
          
          
          Following rules are (at-least) weakly oriented:
             f(a()) =  [9]        
                    >= [11]       
                    =  f(c(a()))  
          
             f(a()) =  [9]        
                    >= [9]        
                    =  f(d(a()))  
          
          f(c(b())) =  [10]       
                    >= [9]        
                    =  f(d(a()))  
          
            f(d(X)) =  [1] X + [8]
                    >= [1] X + [0]
                    =  X          
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: Bounds.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
        - Weak TRS:
            e(g(X)) -> e(X)
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1} / {a/0,b/0,c/1,d/1,g/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,e,f,g}
    + Applied Processor:
        Bounds {initialAutomaton = minimal, enrichment = match}
    + Details:
        The problem is match-bounded by 1.
        The enriched problem is compatible with follwoing automaton.
          a_0() -> 1
          a_1() -> 1
          a_1() -> 3
          b_0() -> 1
          b_1() -> 1
          b_1() -> 3
          c_0(1) -> 1
          c_1(3) -> 2
          d_0(1) -> 1
          d_1(3) -> 2
          e_0(1) -> 1
          f_0(1) -> 1
          f_1(2) -> 1
          g_0(1) -> 1
          3 -> 1
* Step 5: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            e(g(X)) -> e(X)
            f(a()) -> f(c(a()))
            f(a()) -> f(d(a()))
            f(c(X)) -> X
            f(c(a())) -> f(d(b()))
            f(c(b())) -> f(d(a()))
            f(d(X)) -> X
        - Signature:
            {e/1,f/1} / {a/0,b/0,c/1,d/1,g/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c,d,e,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))