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

** Step 1.b:1: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(a()) -> b()
            f(c()) -> d()
            f(g(x,y)) -> g(f(x),f(y))
            f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
            g(x,x) -> h(e(),x)
        - Signature:
            {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h}
    + 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(g) = {1,2},
          uargs(h) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = 0              
          p(b) = 0              
          p(c) = 0              
          p(d) = 0              
          p(e) = 0              
          p(f) = 1 + 3*x1 + x1^2
          p(g) = 1 + x1 + x2    
          p(h) = 1 + x1 + x2    
        
        Following rules are strictly oriented:
           f(a()) = 1                                
                  > 0                                
                  = b()                              
        
           f(c()) = 1                                
                  > 0                                
                  = d()                              
        
        f(g(x,y)) = 5 + 5*x + 2*x*y + x^2 + 5*y + y^2
                  > 3 + 3*x + x^2 + 3*y + y^2        
                  = g(f(x),f(y))                     
        
        
        Following rules are (at-least) weakly oriented:
        f(h(x,y)) =  5 + 5*x + 2*x*y + x^2 + 5*y + y^2
                  >= 5 + 4*x + x^2 + 4*y + y^2        
                  =  g(h(y,f(x)),h(x,f(y)))           
        
           g(x,x) =  1 + 2*x                          
                  >= 1 + x                            
                  =  h(e(),x)                         
        
** Step 1.b:2: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
            g(x,x) -> h(e(),x)
        - Weak TRS:
            f(a()) -> b()
            f(c()) -> d()
            f(g(x,y)) -> g(f(x),f(y))
        - Signature:
            {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h}
    + 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(g) = {1,2},
          uargs(h) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = 0          
          p(b) = 0          
          p(c) = 0          
          p(d) = 0          
          p(e) = 0          
          p(f) = 2*x1^2     
          p(g) = 2 + x1 + x2
          p(h) = 2 + x1 + x2
        
        Following rules are strictly oriented:
        f(h(x,y)) = 8 + 8*x + 4*x*y + 2*x^2 + 8*y + 2*y^2
                  > 6 + x + 2*x^2 + y + 2*y^2            
                  = g(h(y,f(x)),h(x,f(y)))               
        
        
        Following rules are (at-least) weakly oriented:
           f(a()) =  0                                    
                  >= 0                                    
                  =  b()                                  
        
           f(c()) =  0                                    
                  >= 0                                    
                  =  d()                                  
        
        f(g(x,y)) =  8 + 8*x + 4*x*y + 2*x^2 + 8*y + 2*y^2
                  >= 2 + 2*x^2 + 2*y^2                    
                  =  g(f(x),f(y))                         
        
           g(x,x) =  2 + 2*x                              
                  >= 2 + x                                
                  =  h(e(),x)                             
        
** Step 1.b:3: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            g(x,x) -> h(e(),x)
        - Weak TRS:
            f(a()) -> b()
            f(c()) -> d()
            f(g(x,y)) -> g(f(x),f(y))
            f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
        - Signature:
            {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h}
    + 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(g) = {1,2},
          uargs(h) = {2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = 2            
          p(b) = 0            
          p(c) = 2            
          p(d) = 0            
          p(e) = 0            
          p(f) = 2*x1 + 2*x1^2
          p(g) = 2 + x1 + x2  
          p(h) = 1 + x1 + x2  
        
        Following rules are strictly oriented:
        g(x,x) = 2 + 2*x 
               > 1 + x   
               = h(e(),x)
        
        
        Following rules are (at-least) weakly oriented:
           f(a()) =  12                                      
                  >= 0                                       
                  =  b()                                     
        
           f(c()) =  12                                      
                  >= 0                                       
                  =  d()                                     
        
        f(g(x,y)) =  12 + 10*x + 4*x*y + 2*x^2 + 10*y + 2*y^2
                  >= 2 + 2*x + 2*x^2 + 2*y + 2*y^2           
                  =  g(f(x),f(y))                            
        
        f(h(x,y)) =  4 + 6*x + 4*x*y + 2*x^2 + 6*y + 2*y^2   
                  >= 4 + 3*x + 2*x^2 + 3*y + 2*y^2           
                  =  g(h(y,f(x)),h(x,f(y)))                  
        
** Step 1.b:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(a()) -> b()
            f(c()) -> d()
            f(g(x,y)) -> g(f(x),f(y))
            f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
            g(x,x) -> h(e(),x)
        - Signature:
            {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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