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


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WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +(x,y){y -> s(y)} =
            +(x,s(y)) ->^+ s(+(x,y))
              = C[+(x,y) = +(x,y){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(+) = [1] x1 + [0]
            p(0) = [2]         
            p(s) = [1] x1 + [0]
          p(sum) = [1] x1 + [4]
        
        Following rules are strictly oriented:
        sum(0()) = [6]
                 > [2]
                 = 0()
        
        
        Following rules are (at-least) weakly oriented:
         +(x,0()) =  [1] x + [0]   
                  >= [1] x + [0]   
                  =  x             
        
        +(x,s(y)) =  [1] x + [0]   
                  >= [1] x + [0]   
                  =  s(+(x,y))     
        
        sum(s(x)) =  [1] x + [4]   
                  >= [1] x + [4]   
                  =  +(sum(x),s(x))
        
** Step 1.b:2: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(s(x)) -> +(sum(x),s(x))
        - Weak TRS:
            sum(0()) -> 0()
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(+) = x1 + 2*x2    
            p(0) = 0            
            p(s) = 2 + x1       
          p(sum) = 3 + x1 + x1^2
        
        Following rules are strictly oriented:
        +(x,s(y)) = 4 + x + 2*y   
                  > 2 + x + 2*y   
                  = s(+(x,y))     
        
        sum(s(x)) = 9 + 5*x + x^2 
                  > 7 + 3*x + x^2 
                  = +(sum(x),s(x))
        
        
        Following rules are (at-least) weakly oriented:
        +(x,0()) =  x  
                 >= x  
                 =  x  
        
        sum(0()) =  3  
                 >= 0  
                 =  0()
        
** Step 1.b:3: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
        - Weak TRS:
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(+) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(+) = x1 + 2*x2      
            p(0) = 2              
            p(s) = 2 + x1         
          p(sum) = 2 + x1 + 2*x1^2
        
        Following rules are strictly oriented:
        +(x,0()) = 4 + x
                 > x    
                 = x    
        
        
        Following rules are (at-least) weakly oriented:
        +(x,s(y)) =  4 + x + 2*y     
                  >= 2 + x + 2*y     
                  =  s(+(x,y))       
        
         sum(0()) =  12              
                  >= 2               
                  =  0()             
        
        sum(s(x)) =  12 + 9*x + 2*x^2
                  >= 6 + 3*x + 2*x^2 
                  =  +(sum(x),s(x))  
        
** Step 1.b:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
        - Signature:
            {+/2,sum/1} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,sum} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))