* Step 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(:) = 2*x1 + 2*x1*x2 + x2
          p(a) = 0                  
          p(f) = 0                  
          p(g) = 0                  
        
        Following rules are strictly oriented:
        :(+(x,y),z) = 2 + 2*x + 2*x*z + 2*y + 2*y*z + 3*z
                    > 1 + 2*x + 2*x*z + 2*y + 2*y*z + 2*z
                    = +(:(x,z),:(y,z))                   
        
        
        Following rules are (at-least) weakly oriented:
        :(z,+(x,f(y))) =  1 + x + 2*x*z + 4*z                            
                       >= 1 + x                                          
                       =  :(g(z,y),+(x,a()))                             
        
           :(:(x,y),z) =  4*x + 4*x*y + 4*x*y*z + 4*x*z + 2*y + 2*y*z + z
                       >= 2*x + 4*x*y + 4*x*y*z + 2*x*z + 2*y + 2*y*z + z
                       =  :(x,:(y,z))                                    
        
* Step 2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(:(x,y),z) -> :(x,:(y,z))
        - Weak TRS:
            :(+(x,y),z) -> +(:(x,z),:(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 + x1 + x1*x2 + x1^2 + x2
          p(a) = 1                         
          p(f) = 1                         
          p(g) = 0                         
        
        Following rules are strictly oriented:
        :(:(x,y),z) = 3 + 3*x + 5*x*y + x*y*z + 2*x*y^2 + x*z + 4*x^2 + 4*x^2*y + x^2*y^2 + x^2*z + 2*x^3 + 2*x^3*y + x^4 + 3*y + y*z + y^2 + 2*z
                    > 2 + 2*x + x*y + x*y*z + x*y^2 + x*z + x^2 + y + y*z + y^2 + z                                                              
                    = :(x,:(y,z))                                                                                                                
        
        
        Following rules are (at-least) weakly oriented:
        :(z,+(x,f(y))) =  3 + x + x*z + 3*z + z^2                            
                       >= 3 + x                                              
                       =  :(g(z,y),+(x,a()))                                 
        
           :(+(x,y),z) =  3 + 3*x + 2*x*y + x*z + x^2 + 3*y + y*z + y^2 + 2*z
                       >= 3 + x + x*z + x^2 + y + y*z + y^2 + 2*z            
                       =  +(:(x,z),:(y,z))                                   
        
* Step 3: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
        - Weak TRS:
            :(+(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 + 2*x2
          p(a) = 0                        
          p(f) = 1 + x1                   
          p(g) = 0                        
        
        Following rules are strictly oriented:
        :(z,+(x,f(y))) = 5 + 2*x + 2*x*z + 2*y + 2*y*z + 6*z
                       > 3 + 2*x                            
                       = :(g(z,y),+(x,a()))                 
        
        
        Following rules are (at-least) weakly oriented:
        :(+(x,y),z) =  3 + 2*x + 2*x*z + 2*y + 2*y*z + 4*z                  
                    >= 3 + 2*x + 2*x*z + 2*y + 2*y*z + 4*z                  
                    =  +(:(x,z),:(y,z))                                     
        
        :(:(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
                    =  :(x,:(y,z))                                          
        
* Step 4: 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(?,O(n^2))