/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:
            .(x,\(x,y)) -> y
            .(x,e()) -> x
            .(/(y,x),x) -> y
            .(e(),x) -> x
            /(x,x) -> e()
            /(x,\(y,x)) -> y
            /(x,e()) -> x
            /(.(y,x),x) -> y
            \(x,x) -> e()
            \(x,.(x,y)) -> y
            \(/(x,y),x) -> y
            \(e(),x) -> x
        - Signature:
            {./2,//2,\/2} / {e/0}
        - Obligation:
             derivational complexity wrt. signature {.,/,\,e}
    + 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 + [9]
            p(/) = [1] x1 + [1] x2 + [0]
            p(\) = [1] x1 + [1] x2 + [5]
            p(e) = [0]                  
          
          Following rules are strictly oriented:
          .(x,\(x,y)) = [2] x + [1] y + [14]
                      > [1] y + [0]         
                      = y                   
          
             .(x,e()) = [1] x + [9]         
                      > [1] x + [0]         
                      = x                   
          
          .(/(y,x),x) = [2] x + [1] y + [9] 
                      > [1] y + [0]         
                      = y                   
          
             .(e(),x) = [1] x + [9]         
                      > [1] x + [0]         
                      = x                   
          
          /(x,\(y,x)) = [2] x + [1] y + [5] 
                      > [1] y + [0]         
                      = y                   
          
          /(.(y,x),x) = [2] x + [1] y + [9] 
                      > [1] y + [0]         
                      = y                   
          
               \(x,x) = [2] x + [5]         
                      > [0]                 
                      = e()                 
          
          \(x,.(x,y)) = [2] x + [1] y + [14]
                      > [1] y + [0]         
                      = y                   
          
          \(/(x,y),x) = [2] x + [1] y + [5] 
                      > [1] y + [0]         
                      = y                   
          
             \(e(),x) = [1] x + [5]         
                      > [1] x + [0]         
                      = x                   
          
          
          Following rules are (at-least) weakly oriented:
            /(x,x) =  [2] x + [0]
                   >= [0]        
                   =  e()        
          
          /(x,e()) =  [1] x + [0]
                   >= [1] x + [0]
                   =  x          
          
        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:
            /(x,x) -> e()
            /(x,e()) -> x
        - Weak TRS:
            .(x,\(x,y)) -> y
            .(x,e()) -> x
            .(/(y,x),x) -> y
            .(e(),x) -> x
            /(x,\(y,x)) -> y
            /(.(y,x),x) -> y
            \(x,x) -> e()
            \(x,.(x,y)) -> y
            \(/(x,y),x) -> y
            \(e(),x) -> x
        - Signature:
            {./2,//2,\/2} / {e/0}
        - Obligation:
             derivational complexity wrt. signature {.,/,\,e}
    + 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(/) = [1] x1 + [1] x2 + [12]
            p(\) = [1] x1 + [1] x2 + [0] 
            p(e) = [0]                   
          
          Following rules are strictly oriented:
            /(x,x) = [2] x + [12]
                   > [0]         
                   = e()         
          
          /(x,e()) = [1] x + [12]
                   > [1] x + [0] 
                   = x           
          
          
          Following rules are (at-least) weakly oriented:
          .(x,\(x,y)) =  [2] x + [1] y + [0] 
                      >= [1] y + [0]         
                      =  y                   
          
             .(x,e()) =  [1] x + [0]         
                      >= [1] x + [0]         
                      =  x                   
          
          .(/(y,x),x) =  [2] x + [1] y + [12]
                      >= [1] y + [0]         
                      =  y                   
          
             .(e(),x) =  [1] x + [0]         
                      >= [1] x + [0]         
                      =  x                   
          
          /(x,\(y,x)) =  [2] x + [1] y + [12]
                      >= [1] y + [0]         
                      =  y                   
          
          /(.(y,x),x) =  [2] x + [1] y + [12]
                      >= [1] y + [0]         
                      =  y                   
          
               \(x,x) =  [2] x + [0]         
                      >= [0]                 
                      =  e()                 
          
          \(x,.(x,y)) =  [2] x + [1] y + [0] 
                      >= [1] y + [0]         
                      =  y                   
          
          \(/(x,y),x) =  [2] x + [1] y + [12]
                      >= [1] y + [0]         
                      =  y                   
          
             \(e(),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: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            .(x,\(x,y)) -> y
            .(x,e()) -> x
            .(/(y,x),x) -> y
            .(e(),x) -> x
            /(x,x) -> e()
            /(x,\(y,x)) -> y
            /(x,e()) -> x
            /(.(y,x),x) -> y
            \(x,x) -> e()
            \(x,.(x,y)) -> y
            \(/(x,y),x) -> y
            \(e(),x) -> x
        - Signature:
            {./2,//2,\/2} / {e/0}
        - Obligation:
             derivational complexity wrt. signature {.,/,\,e}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))