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

** Step 1.b:1: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            *(x,+(y,z)) -> +(*(x,y),*(x,z))
            *(x,1()) -> x
            *(+(x,y),z) -> +(*(x,z),*(y,z))
            *(1(),y) -> y
        - Signature:
            {*/2} / {+/2,1/0}
        - Obligation:
             runtime complexity wrt. defined symbols {*} and constructors {+,1}
    + 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(+) = {1,2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(*) = 4*x1 + 3*x1*x2 + 5*x2
          p(+) = 2 + x1 + x2          
          p(1) = 2                    
        
        Following rules are strictly oriented:
        *(x,+(y,z)) = 10 + 10*x + 3*x*y + 3*x*z + 5*y + 5*z
                    > 2 + 8*x + 3*x*y + 3*x*z + 5*y + 5*z  
                    = +(*(x,y),*(x,z))                     
        
           *(x,1()) = 10 + 10*x                            
                    > x                                    
                    = x                                    
        
        *(+(x,y),z) = 8 + 4*x + 3*x*z + 4*y + 3*y*z + 11*z 
                    > 2 + 4*x + 3*x*z + 4*y + 3*y*z + 10*z 
                    = +(*(x,z),*(y,z))                     
        
           *(1(),y) = 8 + 11*y                             
                    > y                                    
                    = y                                    
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            *(x,+(y,z)) -> +(*(x,y),*(x,z))
            *(x,1()) -> x
            *(+(x,y),z) -> +(*(x,z),*(y,z))
            *(1(),y) -> y
        - Signature:
            {*/2} / {+/2,1/0}
        - Obligation:
             runtime complexity wrt. defined symbols {*} and constructors {+,1}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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