/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/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:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {minus,quot} 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:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {minus,quot} 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:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {minus,quot} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          minus(x,y){x -> s(x),y -> s(y)} =
            minus(s(x),s(y)) ->^+ minus(x,y)
              = C[minus(x,y) = minus(x,y){}]

** Step 1.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {minus,quot} 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(minus) = {1},
          uargs(quot) = {1},
          uargs(s) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(0) = [1]         
          p(minus) = [1] x1 + [1]
           p(quot) = [4] x1 + [3]
              p(s) = [1] x1 + [2]
        
        Following rules are strictly oriented:
            minus(x,0()) = [1] x + [1]             
                         > [1] x + [0]             
                         = x                       
        
        minus(s(x),s(y)) = [1] x + [3]             
                         > [1] x + [1]             
                         = minus(x,y)              
        
          quot(0(),s(y)) = [7]                     
                         > [1]                     
                         = 0()                     
        
         quot(s(x),s(y)) = [4] x + [11]            
                         > [4] x + [9]             
                         = s(quot(minus(x,y),s(y)))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            minus(x,0()) -> x
            minus(s(x),s(y)) -> minus(x,y)
            quot(0(),s(y)) -> 0()
            quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        - Signature:
            {minus/2,quot/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {minus,quot} 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))