/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sqr,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:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sqr,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:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sqr,sum} 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)) ->^+ +(*(s(x),s(x)),sum(x))
              = C[sum(x) = sum(x){}]

** Step 1.b:1: NaturalPI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(+) = {1,2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(*) = 0          
            p(+) = 1 + x1 + x2
            p(0) = 0          
            p(s) = 1 + x1     
          p(sqr) = 2          
          p(sum) = 8*x1       
        
        Following rules are strictly oriented:
           sqr(x) = 2                     
                  > 0                     
                  = *(x,x)                
        
        sum(s(x)) = 8 + 8*x               
                  > 1 + 8*x               
                  = +(*(s(x),s(x)),sum(x))
        
        sum(s(x)) = 8 + 8*x               
                  > 3 + 8*x               
                  = +(sqr(s(x)),sum(x))   
        
        
        Following rules are (at-least) weakly oriented:
        sum(0()) =  0  
                 >= 0  
                 =  0()
        
** Step 1.b:2: NaturalPI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sum(0()) -> 0()
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(+) = {1,2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(*) = 13     
            p(+) = x1 + x2
            p(0) = 0      
            p(s) = 14 + x1
          p(sqr) = 14     
          p(sum) = 2 + x1 
        
        Following rules are strictly oriented:
        sum(0()) = 2  
                 > 0  
                 = 0()
        
        
        Following rules are (at-least) weakly oriented:
           sqr(x) =  14                    
                  >= 13                    
                  =  *(x,x)                
        
        sum(s(x)) =  16 + x                
                  >= 15 + x                
                  =  +(*(s(x),s(x)),sum(x))
        
        sum(s(x)) =  16 + x                
                  >= 16 + x                
                  =  +(sqr(s(x)),sum(x))   
        
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {sqr,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^1))