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

** Step 1.b:1: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2} / {0/0,1/0,2/0,g/2,i/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {0,1,2,g,i}
    + 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(f) = {2},
          uargs(g) = {1,2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(0) = 1                        
          p(1) = 0                        
          p(2) = 0                        
          p(f) = 1 + 2*x1 + 2*x1*x2 + 2*x2
          p(g) = 2 + x1 + x2              
          p(i) = 2                        
        
        Following rules are strictly oriented:
             f(x,0()) = 3 + 4*x                            
                      > x                                  
                      = x                                  
        
             f(0(),y) = 3 + 4*y                            
                      > y                                  
                      = y                                  
        
        f(1(),g(x,y)) = 5 + 2*x + 2*y                      
                      > x                                  
                      = x                                  
        
        f(2(),g(x,y)) = 5 + 2*x + 2*y                      
                      > y                                  
                      = y                                  
        
          f(g(x,y),z) = 5 + 2*x + 2*x*z + 2*y + 2*y*z + 6*z
                      > 4 + 2*x + 2*x*z + 2*y + 2*y*z + 4*z
                      = g(f(x,z),f(y,z))                   
        
            f(i(x),y) = 5 + 6*y                            
                      > 2                                  
                      = i(x)                               
        
        
        Following rules are (at-least) weakly oriented:
        f(f(x,y),z) =  3 + 4*x + 4*x*y + 4*x*y*z + 4*x*z + 4*y + 4*y*z + 4*z
                    >= 3 + 4*x + 4*x*y + 4*x*y*z + 4*x*z + 4*y + 4*y*z + 4*z
                    =  f(x,f(y,z))                                          
        
** Step 1.b:2: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(f(x,y),z) -> f(x,f(y,z))
        - Weak TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2} / {0/0,1/0,2/0,g/2,i/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {0,1,2,g,i}
    + 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(f) = {2},
          uargs(g) = {1,2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(0) = 0                           
          p(1) = 0                           
          p(2) = 1                           
          p(f) = 1 + x1 + x1*x2 + 2*x1^2 + x2
          p(g) = 1 + x1 + x2                 
          p(i) = 0                           
        
        Following rules are strictly oriented:
        f(f(x,y),z) = 4 + 5*x + 9*x*y + x*y*z + 4*x*y^2 + x*z + 12*x^2 + 12*x^2*y + 2*x^2*y^2 + 2*x^2*z + 8*x^3 + 8*x^3*y + 8*x^4 + 5*y + y*z + 2*y^2 + 2*z
                    > 2 + 2*x + x*y + x*y*z + 2*x*y^2 + x*z + 2*x^2 + y + y*z + 2*y^2 + z                                                                  
                    = f(x,f(y,z))                                                                                                                          
        
        
        Following rules are (at-least) weakly oriented:
             f(x,0()) =  1 + x + 2*x^2                                          
                      >= x                                                      
                      =  x                                                      
        
             f(0(),y) =  1 + y                                                  
                      >= y                                                      
                      =  y                                                      
        
        f(1(),g(x,y)) =  2 + x + y                                              
                      >= x                                                      
                      =  x                                                      
        
        f(2(),g(x,y)) =  6 + 2*x + 2*y                                          
                      >= y                                                      
                      =  y                                                      
        
          f(g(x,y),z) =  4 + 5*x + 4*x*y + x*z + 2*x^2 + 5*y + y*z + 2*y^2 + 2*z
                      >= 3 + x + x*z + 2*x^2 + y + y*z + 2*y^2 + 2*z            
                      =  g(f(x,z),f(y,z))                                       
        
            f(i(x),y) =  1 + y                                                  
                      >= 0                                                      
                      =  i(x)                                                   
        
** Step 1.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2} / {0/0,1/0,2/0,g/2,i/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {0,1,2,g,i}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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