/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^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sum,sum1} 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:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sum,sum1} 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:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          sum(x){x -> s(x)} =
            sum(s(x)) ->^+ +(sum(x),s(x))
              = C[sum(x) = sum(x){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sum,sum1} 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 + [3]
           p(sum) = [9] x1 + [0]
          p(sum1) = [1] x1 + [1]
        
        Following rules are strictly oriented:
         sum(0()) = [18]          
                  > [2]           
                  = 0()           
        
        sum(s(x)) = [9] x + [27]  
                  > [9] x + [0]   
                  = +(sum(x),s(x))
        
        sum1(0()) = [3]           
                  > [2]           
                  = 0()           
        
        
        Following rules are (at-least) weakly oriented:
        sum1(s(x)) =  [1] x + [4]         
                   >= [1] x + [4]         
                   =  s(+(sum1(x),+(x,x)))
        
** Step 1.b:2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Weak TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sum,sum1} 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) = [0]         
             p(s) = [1] x1 + [2]
           p(sum) = [0]         
          p(sum1) = [8] x1 + [0]
        
        Following rules are strictly oriented:
        sum1(s(x)) = [8] x + [16]        
                   > [8] x + [2]         
                   = s(+(sum1(x),+(x,x)))
        
        
        Following rules are (at-least) weakly oriented:
         sum(0()) =  [0]           
                  >= [0]           
                  =  0()           
        
        sum(s(x)) =  [0]           
                  >= [0]           
                  =  +(sum(x),s(x))
        
        sum1(0()) =  [0]           
                  >= [0]           
                  =  0()           
        
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(sum(x),s(x))
            sum1(0()) -> 0()
            sum1(s(x)) -> s(+(sum1(x),+(x,x)))
        - Signature:
            {sum/1,sum1/1} / {+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sum,sum1} 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^1))