/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /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:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a} and constructors {b,c}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a} and constructors {b,c}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {a} and constructors {b,c}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          a(x){x -> b(x)} =
            a(b(x)) ->^+ b(a(x))
              = C[a(x) = a(x){}]

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

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