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

** Step 1.b:1: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2} / {+/2,a/0,f/1,g/2}
        - Obligation:
             runtime complexity wrt. defined symbols {:} and constructors {+,a,f,g}
    + 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},
          uargs(:) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(+) = 1 + x1 + x2            
          p(:) = 1 + 2*x1 + 2*x1*x2 + x2
          p(a) = 0                      
          p(f) = 1 + x1                 
          p(g) = x1                     
        
        Following rules are strictly oriented:
        :(z,+(x,f(y))) = 3 + x + 2*x*z + y + 2*y*z + 6*z                      
                       > 2 + x + 2*x*z + 4*z                                  
                       = :(g(z,y),+(x,a()))                                   
        
           :(:(x,y),z) = 3 + 4*x + 4*x*y + 4*x*y*z + 4*x*z + 2*y + 2*y*z + 3*z
                       > 2 + 4*x + 4*x*y + 4*x*y*z + 2*x*z + 2*y + 2*y*z + z  
                       = :(x,:(y,z))                                          
        
        
        Following rules are (at-least) weakly oriented:
        :(+(x,y),z) =  3 + 2*x + 2*x*z + 2*y + 2*y*z + 3*z
                    >= 3 + 2*x + 2*x*z + 2*y + 2*y*z + 2*z
                    =  +(:(x,z),:(y,z))                   
        
** Step 1.b:2: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            :(+(x,y),z) -> +(:(x,z),:(y,z))
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2} / {+/2,a/0,f/1,g/2}
        - Obligation:
             runtime complexity wrt. defined symbols {:} and constructors {+,a,f,g}
    + 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},
          uargs(:) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(+) = 1 + x1 + x2               
          p(:) = 3*x1 + 2*x1*x2 + x1^2 + x2
          p(a) = 1                         
          p(f) = 1 + x1                    
          p(g) = 0                         
        
        Following rules are strictly oriented:
        :(+(x,y),z) = 4 + 5*x + 2*x*y + 2*x*z + x^2 + 5*y + 2*y*z + y^2 + 3*z
                    > 1 + 3*x + 2*x*z + x^2 + 3*y + 2*y*z + y^2 + 2*z        
                    = +(:(x,z),:(y,z))                                       
        
        
        Following rules are (at-least) weakly oriented:
        :(z,+(x,f(y))) =  2 + x + 2*x*z + y + 2*y*z + 7*z + z^2                                                                                             
                       >= 2 + x                                                                                                                             
                       =  :(g(z,y),+(x,a()))                                                                                                                
        
           :(:(x,y),z) =  9*x + 12*x*y + 4*x*y*z + 4*x*y^2 + 6*x*z + 12*x^2 + 14*x^2*y + 4*x^2*y^2 + 2*x^2*z + 6*x^3 + 4*x^3*y + x^4 + 3*y + 2*y*z + y^2 + z
                       >= 3*x + 6*x*y + 4*x*y*z + 2*x*y^2 + 2*x*z + x^2 + 3*y + 2*y*z + y^2 + z                                                             
                       =  :(x,:(y,z))                                                                                                                       
        
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2} / {+/2,a/0,f/1,g/2}
        - Obligation:
             runtime complexity wrt. defined symbols {:} and constructors {+,a,f,g}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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