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

** Step 1.b:1: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            minus(f(x,y)) -> f(minus(y),minus(x))
            minus(h(x)) -> h(minus(x))
            minus(minus(x)) -> x
        - Signature:
            {minus/1} / {f/2,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus} and constructors {f,h}
    + 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(f) = {1,2},
          uargs(h) = {1}
        
        Following symbols are considered usable:
          {minus}
        TcT has computed the following interpretation:
              p(f) = 8 + x1 + x2
              p(h) = 4 + x1     
          p(minus) = 8 + 2*x1   
        
        Following rules are strictly oriented:
            minus(h(x)) = 16 + 2*x   
                        > 12 + 2*x   
                        = h(minus(x))
        
        minus(minus(x)) = 24 + 4*x   
                        > x          
                        = x          
        
        
        Following rules are (at-least) weakly oriented:
        minus(f(x,y)) =  24 + 2*x + 2*y      
                      >= 24 + 2*x + 2*y      
                      =  f(minus(y),minus(x))
        
** Step 1.b:2: NaturalPI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            minus(f(x,y)) -> f(minus(y),minus(x))
        - Weak TRS:
            minus(h(x)) -> h(minus(x))
            minus(minus(x)) -> x
        - Signature:
            {minus/1} / {f/2,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus} and constructors {f,h}
    + 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(f) = {1,2},
          uargs(h) = {1}
        
        Following symbols are considered usable:
          {minus}
        TcT has computed the following interpretation:
              p(f) = 4 + x1 + x2
              p(h) = 5 + x1     
          p(minus) = 2 + 4*x1   
        
        Following rules are strictly oriented:
        minus(f(x,y)) = 18 + 4*x + 4*y      
                      > 8 + 4*x + 4*y       
                      = f(minus(y),minus(x))
        
        
        Following rules are (at-least) weakly oriented:
            minus(h(x)) =  22 + 4*x   
                        >= 7 + 4*x    
                        =  h(minus(x))
        
        minus(minus(x)) =  10 + 16*x  
                        >= x          
                        =  x          
        
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            minus(f(x,y)) -> f(minus(y),minus(x))
            minus(h(x)) -> h(minus(x))
            minus(minus(x)) -> x
        - Signature:
            {minus/1} / {f/2,h/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {minus} and constructors {f,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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